System and method for monitoring occupancy of a building using a tracer gas concentration monitoring device

ABSTRACT

A building loses or gains heat through its envelope based on the differential between the indoor and outdoor temperatures. The losses or gains are due to conduction and infiltration. Conventionally, these effects are typically estimated by performing an on-site energy audit. However, total thermal conductivity, conduction, and infiltration can be determined empirically. The number of air changes per hour are empirically measured using a tracer gas concentration monitoring device, which enables the infiltration component of total thermal conductivity to be measured directly. The conduction component of thermal conductivity can then be determined by subtracting the infiltration component from the building&#39;s total thermal conductivity.

CROSS-REFERENCE TO RELATED APPLICATION

This patent application is a continuation of U.S. patent applicationSer. No. 15/138,049, filed Apr. 25, 2016, pending, which is acontinuation-in-part of U.S. patent application Ser. No. 14/664,742,filed Mar. 20, 2015, pending, which is a continuation of U.S. patentapplication Ser. No. 14/631,798, filed Feb. 25, 2015, pending, thepriority dates of which are claimed and the disclosures of which areincorporated by reference.

FIELD

This application relates in general to energy conservation and, inparticular, to a system and method for monitoring occupancy of abuilding using a tracer gas concentration monitoring device.

BACKGROUND

The cost of energy has continued to steadily rise as power utilities tryto cope with continually growing demands, increasing fuel prices, andstricter regulatory mandates. Power utilities must maintain existingpower generation and distribution infrastructure, while simultaneouslyfinding ways to add more capacity to meet future needs, both of whichadd to costs. Burgeoning energy consumption continues to impact theenvironment and deplete natural resources.

A major portion of rising energy costs is borne by consumers, who,despite the need, lack the tools and wherewithal to identify the mostcost effective ways to appreciably lower their energy consumption. Forinstance, no-cost behavioral changes, such as adjusting thermostatsettings and turning off unused appliances, and low-cost physicalimprovements, such as switching to energy-efficient light bulbs, may beinsufficient. Moreover, as space heating and air conditioning togetherconsume the most energy in the average home, appreciable decreases inenergy consumption can usually only be achieved by making costlyupgrades to a building's heating or cooling envelope or “shell.”However, identifying those improvements that will yield an acceptablereturn on investment in terms of costs versus energy savings requiresfirst determining building-specific parameters, including thermalconductivity (UA^(Total)) and infiltration.

Heating, ventilating, and air conditioning (HVAC) energy costs aredirectly tied to a building's thermal conductivity. A poorly insulatedhome or a leaky building will require more HVAC usage to maintain adesired interior temperature than would a comparably-sized butwell-insulated and sealed structure. Reducing HVAC energy costs, though,is not as simple as merely choosing a thermostat setting that causes anHVAC system to run for less time or less often. Rather, numerousfactors, including thermal conductivity, HVAC system efficiency, heatingor cooling season durations, and indoor and outdoor temperaturedifferentials all weigh into energy consumption and need be taken intoaccount when seeking an effective yet cost efficient HVAC energysolution.

Conventionally, an on-site energy audit is performed to determine abuilding's thermal conductivity UA^(Total). An energy audit is a laborintensive and intrusive process that involves measuring a building'sphysical dimensions; approximating insulation R-values; detecting airleakage; and estimating infiltration through a blower door test. Anumerical model is run against the audit findings to solve for thermalconductivity. The UA^(Total) is combined with heating and cooling seasondurations and adjusted for HVAC system efficiency, plus any solar ornon-utility supplied power savings fraction. An audit report is thenpresented as a checklist of steps that may be taken to improve thebuilding's shell.

The blower door test part of the audit presents several challenges.Before the test, monitoring equipment must be calibrated on-site tobuilding-specific factors and airtight covers must be placed over allHVAC vents. Exterior doors, windows and other openings must also besealed and a blower door panel will be temporarily placed into anoutside doorway. During the test, a fan in the blower door panel forcesair into or pulls air out of the building to respectively generate apositive or negative pressure differential to the outdoors, and pressuredifferences are measured. Following completion, test results areconverted into pressure values representing normal conditions from whichinfiltration is then estimated.

As an involved process, a blower door test can be costly,time-consuming, and invasive for building owners and occupants.Throughout the test, trained personnel must be on-site. As well, thebuilding is rendered temporarily uninhabitable and must remain closed upfor an extended period of time while a noisy blower fan is run. Inaddition, a blower door test requires specialized equipment and trainedpersonnel, which adds to the cost. Notwithstanding, blower door testresults are fallible and are simply estimates. Calibration errors thatcan invalidate a test can and do occur; moreover, testing results needto be translated from high pressure testing conditions to normativebuilding operating conditions with reliance on an approximation thatprojects infiltration losses.

Therefore, a need remains for a practical model for determining actualand potential energy consumption for the heating and cooling of abuilding.

A further need remains for an approach to estimating structuralinfiltration without the costs and inconvenience of blower door testingmethodologies.

SUMMARY

Fuel consumption for building heating and cooling can be calculatedthrough two practical approaches that characterize a building's thermalefficiency through empirically-measured values and readily-obtainableenergy consumption data, such as available in utility bills, therebyavoiding intrusive and time-consuming analysis with specialized testingequipment. While the discussion is herein centered on building heatingrequirements, the same principles can be applied to an analysis ofbuilding cooling requirements. The first approach can be used tocalculate annual or periodic fuel requirements. The approach requiresevaluating typical monthly utility billing data and approximations ofheating (or cooling) losses and gains.

The second approach can be used to calculate hourly (or interval) fuelrequirements. The approach includes empirically deriving threebuilding-specific parameters: thermal mass, thermal conductivity, andeffective window area. HVAC system power rating and conversion anddelivery efficiency are also parameterized. The parameters are estimatedusing short duration tests that last at most several days. Theparameters and estimated HVAC system efficiency are used to simulate atime series of indoor building temperature. In addition, the secondhourly (or interval) approach can be used to verify or explain theresults from the first annual (or periodic) approach. For instance, timeseries results can be calculated using the second approach over the spanof an entire year and compared to results determined through the firstapproach. Other uses of the two approaches and forms of comparison arepossible.

A building loses or gains heat through its envelope based on thedifferential between the indoor and outdoor temperatures. The losses orgains are due to conduction and infiltration. Conventionally, theseeffects are typically estimated by performing an on-site energy audit.However, total thermal conductivity, conduction, and infiltration can bedetermined empirically. The number of air changes per hour areempirically measured using a CO₂ concentration monitoring device, whichenables the infiltration component of total thermal conductivity to bemeasured directly. The conduction component of thermal conductivity canthen be determined by subtracting the infiltration component from thebuilding's total thermal conductivity.

Baseline CO₂ concentration outside a building under test is chosen.Initial CO₂ concentration inside the building is determined and recordedusing a CO₂ concentration monitoring device. CO₂ concentration isincreased over the initial CO₂ concentration. Sources causing increasein the CO₂ concentration inside the building are negated and thereafterfurther CO₂ concentrations are measured and recorded inside the buildingusing the CO₂ concentration monitoring device until the further CO₂concentrations substantially stabilize. Infiltration of the building isdetermined based on a number of air changes as a function of thedifference of the initial CO₂ concentration less the baseline CO₂concentration over one or more of the further CO₂ concentration at agiven time less the baseline CO₂ concentration and the given time.

One embodiment provides a system and method for monitoring occupancy ofa building using a tracer gas concentration monitoring device. A tracergas concentration monitoring device is provided inside a building undertest and is operable to determine and record an initial tracer gasconcentration, and further operable to measure and record further tracergas concentrations inside the building subsequent to an increase intracer gas concentration over the initial tracer gas concentration and anegation of sources causing the increase in the tracer gas concentrationinside the building until the further tracer gas concentrationsstabilize, and to record additional tracer gas concentrations followingthe stabilization. A storage includes a baseline tracer gasconcentration applicable to outside the building and a total thermalconductivity of the building. A computer processor is provided that isinterfaced to the storage and configured to execute code, the codeincluding: an infiltration module configured to determine infiltrationof the building based on a number of air changes as a function of thedifference of the initial tracer gas concentration less the baselinetracer gas concentration over one or more of the further tracer gasconcentration at a given time less the baseline tracer gas concentrationand the given time; a conduction module configured to determineconduction of the building as the difference of the total thermalconductivity less the infiltration of the building, wherein at least oneimprovement to a shell of the building is performed based on theinfiltration and the conduction; and a monitoring module configured tomonitor occupancy of the building based on the determined infiltrationand the additional tracer gas concentrations.

In a further embodiment, a system and method for controlling ventilationof a building through using a tracer gas concentration monitoring deviceis provided. A tracer gas concentration monitoring device is providedinside a building under test and is operable to determine and record aninitial tracer gas concentration, and further operable to measure andrecord further tracer gas concentrations inside the building subsequentto an increase in tracer gas concentration over the initial tracer gasconcentration and a negation of sources causing the increase in thetracer gas concentration inside the building until the further tracergas concentrations stabilize, and to record additional tracer gasconcentrations following the stabilization. A storage includes abaseline tracer gas concentration applicable to outside the building anda total thermal conductivity of the building. A computer processor isprovided that is interfaced to the storage and configured to executecode, the code including: an infiltration module configured to determineinfiltration of the building based on a number of air changes as afunction of the difference of the initial tracer gas concentration lessthe baseline tracer gas concentration over one or more of the furthertracer gas concentration at a given time less the baseline tracer gasconcentration and the given time; a conduction module configured todetermine conduction of the building as the difference of the totalthermal conductivity less the infiltration of the building, wherein atleast one improvement to a shell of the building is performed based onthe infiltration and the conduction; and a control module configured tocontrol a mechanical ventilation system of the building based on thedetermined infiltration and the additional tracer gas concentrations.

The foregoing approaches, annual (or periodic) and hourly (or interval)improve upon and compliment the standard energy audit-style methodologyof estimating heating (and cooling) fuel consumption in several ways.First, per the first approach, the equation to calculate annual fuelconsumption and its derivatives is simplified over thefully-parameterized form of the equation used in energy audit analysis,yet without loss of accuracy. Second, both approaches require parametersthat can be obtained empirically, rather than from a detailed energyaudit that requires specialized testing equipment and prescribed testconditions. Third, per the second approach, a time series of indoortemperature and fuel consumption data can be accurately generated. Theresulting fuel consumption data can then be used by economic analysistools using prices that are allowed to vary over time to quantifyeconomic impact.

Moreover, the economic value of heating (and cooling) energy savingsassociated with any building shell improvement in any building has beenshown to be independent of building type, age, occupancy, efficiencylevel, usage type, amount of internal electric gains, or amount solargains, provided that fuel has been consumed at some point for auxiliaryheating. The only information required to calculate savings includes thenumber of hours that define the winter season; average indoortemperature; average outdoor temperature; the building's HVAC systemefficiency (or coefficient of performance for heat pump systems); thearea of the existing portion of the building to be upgraded; the R-valueof the new and existing materials; and the average price of energy, thatis, heating fuel.

The CO₂ monitoring device described and applied herein can replace theblower door test component of an on-site energy audit to baselineinfiltration. Use of the device can also be incorporated into themeasurement and evaluation (M&E) portion of a utility's energyefficiency program to verify the effectiveness of building sealinginitiatives.

Still other embodiments will become readily apparent to those skilled inthe art from the following detailed description, wherein are describedembodiments by way of illustrating the best mode contemplated. As willbe realized, other and different embodiments are possible and theembodiments' several details are capable of modifications in variousobvious respects, all without departing from their spirit and the scope.Accordingly, the drawings and detailed description are to be regarded asillustrative in nature and not as restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a functional block diagram showing heating losses and gainsrelative to a structure.

FIG. 2 is a graph showing, by way of example, balance point thermalconductivity.

FIG. 3 is a flow diagram showing a computer-implemented method formodeling periodic building heating energy consumption in accordance withone embodiment.

FIG. 4 is a flow diagram showing a routine for determining heating gainsfor use in the method of FIG. 3.

FIG. 5 is a flow diagram showing a routine for balancing energy for usein the method of FIG. 3.

FIG. 6 is a process flow diagram showing, by way of example, consumerheating energy consumption-related decision points.

FIG. 7 is a table showing, by way of example, data used to calculatethermal conductivity.

FIG. 8 is a table showing, by way of example, thermal conductivityresults for each season using the data in the table of FIG. 7 as inputsinto Equations (29) through (32).

FIG. 9 is a graph showing, by way of example, a plot of the thermalconductivity results in the table of FIG. 8.

FIG. 10 is a graph showing, by way of example, an auxiliary heatingenergy analysis and energy consumption investment options.

FIG. 11 is a functional block diagram showing heating losses and gainsrelative to a structure.

FIG. 12 is a flow diagram showing a computer-implemented method formodeling interval building heating energy consumption in accordance witha further embodiment.

FIG. 13 is a table showing the characteristics of empirical tests usedto solve for the four unknown parameters in Equation (51).

FIG. 14 is a flow diagram showing a routine for empirically determiningbuilding- and equipment-specific parameters using short duration testsfor use in the method of FIG. 12.

FIG. 15 is a graph showing, by way of example, a comparison of auxiliaryheating energy requirements determined by the hourly approach versus theannual approach.

FIG. 16 is a graph showing, by way of example, a comparison of auxiliaryheating energy requirements with the allowable indoor temperaturelimited to 2° F. above desired temperature of 68° F.

FIG. 17 is a graph showing, by way of example, a comparison of auxiliaryheating energy requirements with the size of effective window areatripled from 2.5 m² to 7.5 m².

FIG. 18 is a table showing, by way of example, test data.

FIG. 19 is a table showing, by way of example, the statistics performedon the data in the table of FIG. 18 required to calculate the three testparameters.

FIG. 20 is a graph showing, by way of example, hourly indoor (measuredand simulated) and outdoor (measured) temperatures.

FIG. 21 is a graph showing, by way of example, simulated versus measuredhourly temperature delta (indoor minus outdoor).

FIG. 22 is a flow diagram showing a method for determining infiltrationof a building through empirical testing using a CO₂ concentrationmonitoring device, in accordance with a further embodiment.

FIG. 23 is a graph showing, by way of example, a time series of CO₂concentration levels inside a test house as measured every half-hourfrom Nov. 1 to Dec. 31, 2015.

FIG. 24 is a graph showing, by way of example, the monitored CO₂concentration as measured in locations upstairs and downstairs in thetest house during the two-month period.

FIG. 25 is a graph showing, by way of example, a time series of CO₂concentration levels inside and outside the test house as measured everyhalf-hour from Nov. 1 to Dec. 31, 2015.

FIG. 26 is a graph showing, by way of example, a time series of CO₂concentration levels inside and outside the test house as measured everyhalf-hour for a 24-hour period on Dec. 10, 2015.

FIG. 27 is a graph showing, by way of example, a time series of numbersof ACH for a 24-hour period on Dec. 10, 2015.

FIG. 28 is a graph showing, by way of example, a steady state CO₂concentration levels inside and outside the test house as projected fromNov. 1 to Dec. 31, 2015 for two, four, and six people.

FIG. 29 is a block diagram showing a system for determining infiltrationof a building through empirical testing using a CO₂ concentrationmonitoring device, in accordance with a further embodiment.

DETAILED DESCRIPTION Conventional Energy Audit-Style Approach

Conventionally, estimating periodic HVAC energy consumption andtherefore fuel costs includes analytically determining a building'sthermal conductivity (UA^(Total)) based on results obtained through anon-site energy audit. For instance, J. Randolf and G. Masters, Energyfor Sustainability: Technology, Planning, Policy, pp. 247, 248, 279(2008), present a typical approach to modeling heating energyconsumption for a building, as summarized therein by Equations 6.23,6.27, and 7.5. The combination of these equations states that annualheating fuel consumption Q^(Fuel) equals the product of UA^(Total), 24hours per day, and the number of heating degree days (HDD) associatedwith a particular balance point temperature T^(Balance Point), asadjusted for the solar savings fraction (SSF) (or non-utility suppliedpower savings fraction) divided by HVAC system efficiency (η^(HVAC)):

$\begin{matrix}{Q^{Fuel} = {( {UA}^{Total} )( {24*{HDD}^{T^{{Balance}\mspace{14mu} {Point}}}} )( {1 - {SSF}} )( \frac{1}{\eta^{HVAC}} )}} & (1)\end{matrix}$

such that:

$\begin{matrix}{T^{{Balance}\mspace{14mu} {Point}} = {T^{{Set}\mspace{14mu} {Point}} - \frac{{Internal}\mspace{14mu} {Gains}}{{UA}^{Total}}}} & (2) \\{and} & \; \\{\eta^{HVAC} = {\eta^{Furnace}\eta^{Distribution}}} & (3)\end{matrix}$

where T^(Set Point) represents the temperature setting of thethermostat, Internal Gains represents the heating gains experiencedwithin the building as a function of heat generated by internal sourcesand auxiliary heating, as further discussed infra, η^(Furnace)represents the efficiency of the furnace or heat source proper, andη^(Distribution) represents the efficiency of the duct work and heatdistribution system. For clarity, HDD^(T) ^(Balance Point) will beabbreviated to HDD^(Balance Point Temp).

A cursory inspection of Equation (1) implies that annual fuelconsumption is linearly related to a building's thermal conductivity.This implication further suggests that calculating fuel savingsassociated with building envelope or shell improvements isstraightforward. In practice, however, such calculations are notstraightforward because Equation (1) was formulated with the goal ofdetermining the fuel required to satisfy heating energy needs. As such,there are several additional factors that the equation must take intoconsideration.

First, Equation (1) needs to reflect the fuel that is required only whenindoor temperature exceeds outdoor temperature. This need led to theheating degree day (HDD) approach (or could be applied on a shorter timeinterval basis of less than one day) of calculating the differencebetween the average daily (or hourly) indoor and outdoor temperaturesand retaining only the positive values. This approach complicatesEquation (1) because the results of a non-linear term must be summed,that is, the maximum of the difference between average indoor andoutdoor temperatures and zero. Non-linear equations complicateintegration, that is, the continuous version of summation.

Second, Equation (1) includes the term Balance Point temperature(T^(Balance Point)). The goal of including the term T^(Balance Point)was to recognize that the internal heating gains of the buildingeffectively lowered the number of degrees of temperature that auxiliaryheating needed to supply relative to the temperature setting of thethermostat T^(Set Point). A balance point temperature T^(Balance Point)of 65° F. was initially selected under the assumption that 65° F.approximately accounted for the internal gains. As buildings became moreefficient, however, an adjustment to the balance point temperatureT^(Balance Point) was needed based on the building's thermalconductivity (UA^(Total)) and internal gains. This assumption furthercomplicated Equation (1) because the equation became indirectlydependent on (and inversely related to) UA^(Total) throughT^(Balance Point).

Third, Equation (1) addresses fuel consumption by auxiliary heatingsources. As a result, Equation (1) must be adjusted to account for solargains. This adjustment was accomplished using the Solar Savings Fraction(SSF). The SSF is based on the Load Collector Ratio (see Eq. 7.4 inRandolf and Masters, p. 278, cited supra, for information about theLCR). The LCR, however, is also a function of UA^(Total). As a result,the SSF is a function of UA^(Total) in a complicated, non-closed formsolution manner. Thus, the SSF further complicates calculating the fuelsavings associated with building shell improvements because the SSF isindirectly dependent on UA^(Total).

As a result, these direct and indirect dependencies in Equation (1)significantly complicate calculating a change in annual fuel consumptionbased on a change in thermal conductivity. The difficulty is madeevident by taking the derivative of Equation (1) with respect to achange in thermal conductivity. The chain and product rules fromcalculus need to be employed since HDD^(Balance Point Temp) and SSF areindirectly dependent on UA^(Total):

$\begin{matrix}{\frac{{dQ}^{Fuel}}{{dUA}^{Total}} = {\{ {{( {UA}^{Total} )\lbrack {{( {HDD}^{{Balance}\mspace{14mu} {Point}\mspace{14mu} {Temp}} )( {{- \frac{dSSF}{dLCR}}\frac{dLCR}{{dUA}^{Total}}} )} + {( {\frac{{dHDD}^{{Balance}\mspace{14mu} {Point}\mspace{14mu} {Temp}}}{{dT}^{{Balance}\mspace{14mu} {Point}}}\frac{{dT}^{{Balance}\mspace{14mu} {Point}}}{{dUA}^{Total}}} )( {1 - {SSF}} )}} \rbrack} + {( {HDD}^{{Balance}\mspace{14mu} {Point}\mspace{14mu} {Temp}} )( {1 - {SSF}} )}} \} ( \frac{24}{\eta^{HVAC}} )}} & (4)\end{matrix}$

The result is Equation (4), which is an equation that is difficult tosolve due to the number and variety of unknown inputs that are required.

To add even further complexity to the problem of solving Equation (4),conventionally, UA^(Total) is determined analytically by performing adetailed energy audit of a building. An energy audit involves measuringphysical dimensions of walls, windows, doors, and other building parts;approximating R-values for thermal resistance; estimating infiltrationusing a blower door test; and detecting air leakage. A numerical modelis then run to perform the calculations necessary to estimate totalthermal conductivity. Such an energy audit can be costly, timeconsuming, and invasive for building owners and occupants. Moreover, asa calculated result, the value estimated for UA^(Total) carries thepotential for inaccuracies, as the model is strongly influenced byphysical mismeasurements or omissions, data assumptions, and so forth.

Empirically-Based Approaches to Modeling Heating Fuel Consumption

A building loses or gains heat through its envelope based on thedifferential between the indoor and outdoor temperatures. The losses orgains are due to conduction and infiltration. Conventionally, theseeffects are typically estimated by performing an on-site energy audit.However, total thermal conductivity, conduction, and infiltration can bedetermined empirically. In one embodiment, building heating (andcooling) fuel consumption can be calculated through empirical twoapproaches, annual (or periodic) and hourly (or interval), to thermallycharacterize a building without intrusive and time-consuming tests. Thefirst approach, referred to as a Virtual Energy Audit, as furtherdescribed infra beginning with reference to BRIEF DESCRIPTION OF THEDRAWINGS

FIG. 1, can be performed without placing any equipment on-site and onlyrequires typical monthly utility billing data and approximations ofheating (or cooling) losses and gains. The second approach, referred toas a Lean Energy Audit, as further described infra beginning withreference to FIG. 11, requires placing minimal monitoring equipmenton-site and involves empirically deriving three building-specificparameters, thermal mass, thermal conductivity, and effective windowarea, plus HVAC system efficiency using short duration tests that lastat most several days. The three building-specific parameters thusobtained can then be used to simulate a time series of indoor buildingtemperature, seasonal fuel consumption, and maximum indoor temperature.

The Virtual Energy Audit and Lean Energy Audit approaches empiricallymeasure total thermal conductivity UA^(Total). As described, forinstance, in commonly-assigned U.S. patent application, entitled“Computer-Implemented System and Method for Interactively EvaluatingPersonal Energy-Related Investments,” Ser. No. 14/294,079, filed Jun. 2,2014, pending, the disclosure of which is incorporated by reference,UA^(Total) equals heat loss or gain due to conduction plus infiltration,which can be formulaically expressed as:

$\begin{matrix}{{UA}^{Total} = {\overset{\overset{Conduction}{}}{( {\sum\limits_{i = 1}^{N}{U^{i}A^{i}}} )} + \overset{\overset{Infitration}{}}{\rho \; c\; {nV}}}} & (5)\end{matrix}$

where U^(i) represents the inverse of the R-value and A^(i) representsthe area of surface i; ρ is a constant that represents the density ofair (lbs./ft³); cis a constant that represents the specific heat of air(Btu/lb.-° F.); n is the number of air changes per hour (ACH); and Vrepresents the volume of air per air change (ft³/AC). The density of airρ and the specific heat of air c are the same for all buildings andrespectively equal 0.075 lbs./ft³ and 0.24 Btu/lb.-° F. The number ofACH n and the volume of air per air change V are building-specificvalues. Volume V can be measured directly or can be approximated bymultiplying building square footage times the average room height.Number of ACH n can be estimated using a blower door test, such asdescribed supra. Alternatively, in a further embodiment, the number ofACH n can be empirically measured under actual operating conditions, asfurther described infra beginning with reference to FIG. 22, whichenables the infiltration component of total thermal conductivity to bemeasured directly. The conduction component of thermal conductivity canthen be determined by subtracting the infiltration component from thebuilding's total thermal conductivity.

While the discussion herein is centered on building heatingrequirements, the same principles can be applied to an analysis ofbuilding cooling requirements. In addition, conversion factors foroccupant heating gains (250 Btu of heat per person per hour), heatinggains from internal electricity consumption (3,412 Btu per kWh), solarresource heating gains (3,412 Btu per kWh), and fuel pricing

$( \frac{{Price}^{NG}}{10^{5}} $

if in units of $ per therm and

$\frac{{Price}^{Electricity}}{3,412}$

if in units of $ per kWh) are used by way of example; other conversionfactors or expressions are possible.

First Approach: Annual (or Periodic) Fuel Consumption

Fundamentally, thermal conductivity is the property of a material, here,a structure, to conduct heat. 1-72—is a functional block diagram 10showing heating losses and gains relative to a structure 11.Inefficiencies in the shell 12 (or envelope) of a structure 11 canresult in losses in interior heating 14, whereas gains 13 in heatinggenerally originate either from sources within (or internal to) thestructure 11, including heating gains from occupants 15, gains fromoperation of electric devices 16, and solar gains 17, or from auxiliaryheating sources 18 that are specifically intended to provide heat to thestructure's interior.

In this first approach, the concepts of balance point temperatures andsolar savings fractions, per Equation (1), are eliminated. Instead,balance point temperatures and solar savings fractions are replaced withthe single concept of balance point thermal conductivity. Thissubstitution is made by separately allocating the total thermalconductivity of a building (UA^(Total)) to thermal conductivity forinternal heating gains (UA^(Balance Point)), including occupancy, heatproduced by operation of certain electric devices, and solar gains, andthermal conductivity for auxiliary heating (UA^(Auxiliary Heating)). Theend result is Equation (35), further discussed in detail infra, whicheliminates the indirect and non-linear parameter relationships inEquation (1) to UA^(Total).

The conceptual relationships embodied in Equation (35) can be describedwith the assistance of a diagram. FIG. 2 is a graph 20 showing, by wayof example, balance point thermal conductivity UA^(Balance Point), thatis, the thermal conductivity for internal heating gains. The x-axis 21represents total thermal conductivity, UA^(Total), of a building (inunits of Btu/hr-° F.). The y-axis 22 represents total heating energyconsumed to heat the building. Total thermal conductivity 21 (along thex-axis) is divided into “balance point” thermal conductivity(UA^(Balance Point)) 23 and “heating system” (or auxiliary heating)thermal conductivity (UA^(Auxiliary Heating)) 24. “Balance point”thermal conductivity 23 characterizes heating losses, which can occur,for example, due to the escape of heat through the building envelope tothe outside and by the infiltration of cold air through the buildingenvelope into the building's interior that are compensated for byinternal gains. “Heating system” thermal conductivity 24 characterizesheating gains, which reflects the heating delivered to the building'sinterior above the balance point temperature T^(Balance Point),generally as determined by the setting of the auxiliary heating source'sthermostat or other control point.

In this approach, total heating energy 22 (along the y-axis) is dividedinto gains from internal heating 25 and gains from auxiliary heatingenergy 25. Internal heating gains are broken down into heating gainsfrom occupants 27, gains from operation of electric devices 28 in thebuilding, and solar gains 29. Sources of auxiliary heating energyinclude, for instance, natural gas furnace 30 (here, with a 56%efficiency), electric resistance heating 31 (here, with a 100%efficiency), and electric heat pump 32 (here, with a 250% efficiency).Other sources of heating losses and gains are possible.

The first approach provides an estimate of fuel consumption over a yearor other period of inquiry based on the separation of thermalconductivity into internal heating gains and auxiliary heating. FIG. 3is a flow diagram showing a computer-implemented method 40 for modelingperiodic building heating energy consumption in accordance with oneembodiment. Execution of the software can be performed with theassistance of a computer system, such as further described infra withreference to FIG. 29, as a series of process or method modules or steps.

In the first part of the approach (steps 41-43), heating losses andheating gains are separately analyzed. In the second part of theapproach (steps 44-46), the portion of the heating gains that need to beprovided by fuel, that is, through the consumption of energy forgenerating heating using auxiliary heating 18 (shown in 1-72-), isdetermined to yield a value for annual (or periodic) fuel consumption.Each of the steps will now be described in detail.

Specify Time Period

Heating requirements are concentrated during the winter months, so as aninitial step, the time period of inquiry is specified (step 41). Theheating degree day approach (HDD) in Equation (1) requires examining allof the days of the year and including only those days where outdoortemperatures are less than a certain balance point temperature. However,this approach specifies the time period of inquiry as the winter seasonand considers all of the days (or all of the hours, or other time units)during the winter season. Other periods of inquiry are also possible,such as a five- or ten-year time frame, as well as shorter time periods,such as one- or two-month intervals.

Separate Heating Losses from Heating Gains

Heating losses are considered separately from heating gains (step 42).The rationale for drawing this distinction will now be discussed.

Heating Losses

For the sake of discussion herein, those regions located mainly in thelower latitudes, where outdoor temperatures remain fairly moderate yearround, will be ignored and focus placed instead on those regions thatexperience seasonal shifts of weather and climate. Under thisassumption, a heating degree day (HDD) approach specifies that outdoortemperature must be less than indoor temperature. No such limitation isapplied in this present approach. Heating losses are negative if outdoortemperature exceeds indoor temperature, which indicates that thebuilding will gain heat during these times. Since the time period hasbeen limited to only the winter season, there will likely to be alimited number of days when that situation could occur and, in thoselimited times, the building will benefit by positive heating gain. (Notethat an adjustment would be required if the building took advantage ofthe benefit of higher outdoor temperatures by circulating outdoor airinside when this condition occurs. This adjustment could be made bytreating the condition as an additional source of heating gain.)

As a result, fuel consumption for heating losses Q^(Losses) over thewinter season equals the product of the building's total thermalconductivity UA^(Total) and the difference between the indoor T^(Indoor)and outdoor temperature T^(Outdoor), summed over all of the hours of thewinter season:

$\begin{matrix}{Q^{Losses} = {\sum\limits_{t^{Start}}^{t^{End}}{( {UA}^{Total} )( {T_{t}^{Indoor} - T_{t}^{Outdoor}} )}}} & (6)\end{matrix}$

where Start and End respectively represent the first and last hours ofthe winter (heating) season.

Equation (6) can be simplified by solving the summation. Thus, totalheating losses Q^(Losses) equal the product of thermal conductivityUA^(Total) and the difference between average indoor temperature T^(Indoor) and average outdoor temperature T ^(Outdoor) over the winterseason and the number of hours H in the season over which the average iscalculated:

Q _(Losses)=(UA ^(Total))( T ^(Indoor) −T ^(Outdoor))(H)  (7)

Heating Gains

Heating gains are calculated for two broad categories (step 43) based onthe source of heating, internal heating gains Q^(Gains-Internal) andauxiliary heating gains Q^(Gains-Auxiliary Heating), as furtherdescribed infra with reference to FIG. 4. Internal heating gains can besubdivided into heating gained from occupants Q^(Gains-Occupants),heating gained from the operation of electric devicesQ^(Gains-Electric), and heating gained from solar heatingQ^(Gains-Solar). Other sources of internal heating gains are possible.The total amount of heating gained Q^(Gains) from these two categoriesof heating sources equals:

Q ^(Gains) =Q ^(Gains-Internal) +Q ^(Gains-Auxiliary Heating)  (8)

where

Q ^(Gains-Internal) =Q ^(Gains-Occupants) +Q ^(Gains-Electric) +Q^(Gains-Solar)  (9)

Calculate Heating Gains

Equation (9) states that internal heating gains Q^(Gains-Internal)include heating gains from Occupant, Electric, and Solar heatingsources. FIG. 4 is a flow diagram showing a routine 50 for determiningheating gains for use in the method 40 of FIG. 3 Each of these heatinggain sources will now be discussed.

Occupant Heating Gains

People occupying a building generate heat. Occupant heating gainsQ^(Gains-Occupants) (step 51) equal the product of the heat produced perperson, the average number of people in a building over the time period,and the number of hours (H) (or other time units) in that time period.Let P represent the average number of people. For instance, using aconversion factor of 250 Btu of heat per person per hour, heating gainsfrom the occupants Q^(Gains-Occupants) equal:

Q ^(Gains-Occupants)=250( P )(H)  (10)

Other conversion factors or expressions are possible.

Electric Heating Gains

The operation of electric devices that deliver all heat that isgenerated into the interior of the building, for instance, lights,refrigerators, and the like, contribute to internal heating gain.Electric heating gains Q^(Gains-Electric) (step 52) equal the amount ofelectricity used in the building that is converted to heat over the timeperiod.

Care needs to be taken to ensure that the measured electricityconsumption corresponds to the indoor usage. Two adjustments may berequired. First, many electric utilities measure net electricityconsumption. The energy produced by any photovoltaic (PV) system needsto be added back to net energy consumption (Net) to result in grossconsumption if the building has a net-metered PV system. This amount canbe estimated using time- and location-correlated solar resource data, aswell as specific information about the orientation and othercharacteristics of the photovoltaic system, such as can be provided bythe Solar Anywhere SystemCheck service (http://www.SolarAnywhere.com), aWeb-based service operated by Clean Power Research, L.L.C., Napa,Calif., with the approach described, for instance, in commonly-assignedU.S. patent application, entitled “Computer-Implemented System andMethod for Estimating Gross Energy Load of a Building,” Ser. No.14/531,940, filed Nov. 3, 2014, pending, the disclosure of which isincorporated by reference, or measured directly.

Second, some uses of electricity may not contribute heat to the interiorof the building and need be factored out as external electric heatinggains (External). These uses include electricity used for electricvehicle charging, electric dryers (assuming that most of the hot exhaustair is vented outside of the building, as typically required by buildingcode), outdoor pool pumps, and electric water heating using eitherdirect heating or heat pump technologies (assuming that most of the hotwater goes down the drain and outside the building—a large body ofstanding hot water, such as a bathtub filled with hot water, can beconsidered transient and not likely to appreciably increase thetemperature indoors over the long run).

For instance, using a conversion factor from kWh to Btu of 3,412 Btu perkWh (since Q^(Gains-Electric) is in units of Btu), internal electricgains Q^(Gains-Electric) equal:

$\begin{matrix}{Q^{{Gains} - {Electric}} = {( \overset{\_}{{Net} + {PV} - {External}} )(H)( \frac{3,412\mspace{14mu} {Btu}}{kWh} )}} & (11)\end{matrix}$

where Net represents net energy consumption, PV represents any energyproduced by a PV system, External represents heating gains attributableto electric sources that do not contribute heat to the interior of abuilding. Other conversion factors or expressions are possible. Theaverage delivered electricity Net+PV−External equals the total over thetime period divided by the number of hours (H) in that time period.

$\begin{matrix}{\overset{\_}{{Net} + {PV} - {External}} = \frac{{Net} + {PV} - {External}}{H}} & (12)\end{matrix}$

Solar Heating Gains

Solar energy that enters through windows, doors, and other openings in abuilding as sunlight will heat the interior. Solar heating gainsQ^(Gains-Solar) (step 53) equal the amount of heat delivered to abuilding from the sun. In the northern hemisphere, Q^(Gains-Solar) canbe estimated based on the south-facing window area (m²) times the solarheating gain coefficient (SHGC) times a shading factor; together, theseterms are represented by the effective window area (W). Solar heatinggains Q^(Gains-Solar) equal the product of W, the average directvertical irradiance (DVI) available on a south-facing surface (Solar, asrepresented by DVI in kW/m²), and the number of hours (H) in the timeperiod. For instance, using a conversion factor from kWh to Btu of 3,412Btu per kWh (since Q^(Gains-Solar) is in units of Btu while averagesolar is in kW/m²), solar heating gains Q^(Gains-Solar) equal:

$\begin{matrix}{Q^{{Gains} - {Solar}} = {( \overset{\_}{Solar} )(W)(H)( \frac{3,412\mspace{14mu} {Btu}}{kWh} )}} & (13)\end{matrix}$

Other conversion factors or expressions are possible.

Note that for reference purposes, the SHGC for one particular highquality window designed for solar gains, the Andersen High-PerformanceLow-E4 PassiveSun Glass window product, manufactured by AndersenCorporation, Bayport, Minn., is 0.54; many windows have SHGCs that arebetween 0.20 to 0.25.

Auxiliary Heating Gains

The internal sources of heating gain share the common characteristic ofnot being operated for the sole purpose of heating a building, yetnevertheless making some measureable contribution to the heat to theinterior of a building. The fourth type of heating gain, auxiliaryheating gains Q^(Gains-Auxiliary Heating), consumes fuel specifically toprovide heat to the building's interior and, as a result, must includeconversion efficiency. The gains from auxiliary heating gainsQ^(Gains-Auxiliary Heating) (step 53) equal the product of the averagehourly fuel consumed Q ^(Fuel) times the hours (H) in the period timesHVAC system efficiency η^(HVAC).

Q ^(Gains-Auxiliary Heating)=( Q ^(Fuel))(H)(η^(HVAC))  (14)

Equation (14) can be stated in a more general form that can be appliedto both heating and cooling seasons by adding a binary multiplier,HeatOrCool. The binary multiplier HeatOrCool equals 1 when the heatingsystem is in operation and equals −1 when the cooling system is inoperation. This more general form will be used in a subsequent section.

Q ^(Gains(Losses)-HVAC)=(HeatOrCool)( Q ^(Fuel))(H)(η^(HVAC))  (15)

Divide Thermal Conductivity into Parts

Consider the situation when the heating system is in operation. TheHeatingOrCooling term in Equation (15) equals 1 in the heating season.As illustrated in FIG. 3, a building's thermal conductivity UA^(Total),rather than being treated as a single value, can be conceptually dividedinto two parts (step 44), with a portion of UA^(Total) allocated to“balance point thermal conductivity” (UA^(Balance Point)) and a portionto “auxiliary heating thermal conductivity” (UA^(Auxiliary Heating)),such as pictorially described supra with reference to FIG. 2.UA^(Balance Point) corresponds to the heating losses that a building cansustain using only internal heating gains Q^(Gains-Internal). This valueis related to the concept that a building can sustain a specifiedbalance point temperature in light of internal gains. However, insteadof having a balance point temperature, some portion of the buildingUA^(Balance Point) is considered to be thermally sustainable givenheating gains from internal heating sources (Q^(Gains-Internal)). As therest of the heating losses must be made up by auxiliary heating gains,the remaining portion of the building UA^(Auxiliary Heating) isconsidered to be thermally sustainable given heating gains fromauxiliary heating sources (Q^(Gains-Auxiliary Heating)). The amount ofauxiliary heating gained is determined by the setting of the auxiliaryheating source's thermostat or other control point. Thus, UA^(Total) canbe expressed as:

UA ^(Total) =UA ^(Balance Point) +UA ^(Auxiliary Heating)  (16)

where

UA ^(Balance Point) =UA ^(Occupants) +UA ^(Electric) +UA ^(Solar)  (17)

such that UA^(Occupant), UA^(Electric), and UA^(Solar) respectivelyrepresent the thermal conductivity of internal heating sources,specifically, occupants, electric and solar.

In Equation (16), total thermal conductivity UA^(Total) is fixed at acertain value for a building and is independent of weather conditions;UA^(Total) depends upon the building's efficiency. The component partsof Equation (16), balance point thermal conductivity UA^(Balance Point)and auxiliary heating thermal conductivity UA^(Auxiliary Heating),however, are allowed to vary with weather conditions. For example, whenthe weather is warm, there may be no auxiliary heating in use and all ofthe thermal conductivity will be allocated to the balance point thermalconductivity UA^(Balance Point) component.

Fuel consumption for heating losses Q^(Losses) can be determined bysubstituting Equation (16) into Equation (7):

Q ^(Losses)=(UA ^(Balance Point) +UA ^(Auxiliary Heating))( T ^(Indoor)−T ^(Outdoor))(H)  (18)

Balance Energy

Heating gains must equal heating losses for the system to balance (step45), as further described infra with reference to FIG. 5. Heating energybalance is represented by setting Equation (8) equal to Equation (18):

$\begin{matrix}{{Q^{{Gains} - {Internal}} + Q^{{Gains} - {{Auxiliary}\mspace{14mu} {Heating}}}} = {( {{UA}^{{Balance}\mspace{14mu} {Point}} + {UA}^{{Auxiliary}\mspace{14mu} {Heating}}} )( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}} & (19)\end{matrix}$

The result can then be divided by (T ^(Indoor)−T ^(Outdoor))(H),assuming that this term is non-zero:

$\begin{matrix}{{{UA}^{{Balance}\mspace{14mu} {Point}} + {UA}^{{Auxiliary}\mspace{14mu} {Heating}}} = \frac{Q^{{Gains} - {Internal}} + Q^{{Gains} - {{Auxiliary}\mspace{14mu} {Heating}}}}{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}} & (20)\end{matrix}$

Equation (20) expresses energy balance as a combination of bothUA^(Balance Point) and UA^(Auxiliary Heating). FIG. 5 is a flow diagramshowing a routine 60 for balancing energy for use in the method 40 ofFIG. 3. Equation (20) can be further constrained by requiring that thecorresponding terms on each side of the equation match, which willdivide Equation (20) into a set of two equations:

$\begin{matrix}{{UA}^{{Balance}\mspace{14mu} {Point}} = \frac{Q^{{Gains} - {Internal}}}{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}} & (21) \\{{UA}^{{Auxiliary}\mspace{14mu} {Heating}} = \frac{Q^{{Gains} - {{Auxiliary}\mspace{14mu} {Heating}}}}{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}} & (22)\end{matrix}$

The UA^(Balance Point) should always be a positive value. Equation (21)accomplishes this goal in the heating season. An additional term,HeatOrCool is required for the cooling season that equals 1 in theheating season and −1 in the cooling season.

$\begin{matrix}{{UA}^{{Balance}\mspace{14mu} {Point}} = \frac{({HeatOrCool})( Q^{{Gains} - {Internal}} )}{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}} & (23)\end{matrix}$

HeatOrCool and its inverse are the same. Thus, internal gains equals:

Q ^(Gains-Internal)=(HeatOrCool)(UA ^(Balance Point))( T ^(Indoor) −T^(Outdoor))(H)  (24)

Components of UA^(Balance Point)

For clarity, UA^(Balance Point) can be divided into three componentvalues (step 61) by substituting Equation (9) into Equation (21):

$\begin{matrix}{{UA}^{{Balance}\mspace{14mu} {Point}} = \frac{Q^{{Gains} - {Occupants}} + Q^{{Gains} - {Electric}} + Q^{{Gains} - {Solar}}}{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}} & (25)\end{matrix}$

Since UA^(Balance Point) equals the sum of three component values (asspecified in Equation (17)), Equation (25) can be mathematically limitedby dividing Equation (25) into three equations:

$\begin{matrix}{{UA}^{Occupants} = \frac{Q^{{Gains} - {Occupants}}}{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}} & (26) \\{{UA}^{Electric} = \frac{Q^{{Gains} - {Electric}}}{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}} & (27) \\{{UA}^{Solar} = \frac{Q^{{Gains} - {Solar}}}{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}} & (28)\end{matrix}$

Solutions for Components of UA^(Balance Point) andUA^(Auxiliary Heating)

The preceding equations can be combined to present a set of results withsolutions provided for the four thermal conductivity components asfollows. First, the portion of the balance point thermal conductivityassociated with occupants UA^(Occupants) (step 62) is calculated bysubstituting Equation (10) into Equation (26). Next, the portion of thebalance point thermal conductivity UA^(Electric) associated withinternal electricity consumption (step 63) is calculated by substitutingEquation (11) into Equation (27). Internal electricity consumption isthe amount of electricity consumed internally in the building andexcludes electricity consumed for HVAC operation, pool pump operation,electric water heating, electric vehicle charging, and so on, sincethese sources of electricity consumption result in heat or work beingused external to the inside of the building. The portion of the balancepoint thermal conductivity UA^(Solar) associated with solar gains (step64) is then calculated by substituting Equation (13) into Equation (28).Finally, thermal conductivity UA^(Auxiliary Heating) associated withauxiliary heating (step 64) is calculated by substituting Equation (14)into Equation (22).

$\begin{matrix}{{UA}^{Occupants} = \frac{250( \overset{\_}{P} )}{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}} & (29) \\{{UA}^{Electric} = {\frac{( \overset{\_}{{Net} + {PV} - {External}} )}{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}( \frac{3\text{,}412\mspace{14mu} {Btu}}{kWh} )}} & (30) \\{{UA}^{Solar} = {\frac{( \overset{\_}{Solar} )(W)}{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}( \frac{3\text{,}412\mspace{14mu} {Btu}}{kWh} )}} & (31) \\{{UA}^{{Auxiliary}\mspace{14mu} {Heating}} = \frac{{\overset{\_}{Q}}^{Fuel}\eta^{HVAC}}{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}} & (32)\end{matrix}$

Determine Fuel Consumption

Referring back to FIG. 3, Equation (32) can used to derive a solution toannual (or periodic) heating fuel consumption. First, Equation (16) issolved for UA^(Auxiliary Heating):

UA ^(Auxiliary Heating) =UA ^(Total) −UA ^(Balance Point)  (33)

Equation (33) is then substituted into Equation (32):

$\begin{matrix}{{{UA}^{Total} - {UA}^{{Balance}\mspace{14mu} {Point}}} = \frac{{\overset{\_}{Q}}^{Fuel}\eta^{HVAC}}{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}} & (34)\end{matrix}$

Finally, solving Equation (34) for fuel and multiplying by the number ofhours (H) in (or duration of) the time period yields:

$\begin{matrix}{Q^{Fuel} = \frac{( {{UA}^{Total} - {UA}^{{Balance}\mspace{14mu} {Point}}} )( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}{\eta^{HVAC}}} & (35)\end{matrix}$

Equation (35) is valid during the heating season and applies whereUA^(Total)≥UA^(Balance Point). Otherwise, fuel consumption is 0.

Using Equation (35), annual (or periodic) heating fuel consumptionQ^(Fuel) can be determined (step 46). The building's thermalconductivity UA^(Total), if already available through, for instance, theresults of an energy audit, is obtained. Otherwise, UA^(Total) can bedetermined by solving Equations (29) through (32) using historical fuelconsumption data, such as shown, by way of example, in the table of FIG.7, or by solving Equation (53), as further described infra. UA^(Total)can also be empirically determined with the approach described, forinstance, in commonly-assigned U.S. Pat. No. 10,024,733, issued Jul. 17,2018, the disclosure of which is incorporated by reference. Other waysto determine UA^(Total) are possible. UA^(Balance Point) can bedetermined by solving Equation (25). The remaining values, averageindoor temperature T ^(Indoor) and average outdoor temperature T^(Outdoor), and HVAC system efficiency η^(HVAC), can respectively beobtained from historical weather data and manufacturer specifications.

Practical Considerations

Equation (35) is empowering. Annual heating fuel consumption Q^(Fuel)can be readily determined without encountering the complications ofEquation (1), which is an equation that is difficult to solve due to thenumber and variety of unknown inputs that are required. The implicationsof Equation (35) in consumer decision-making, a general discussion, andsample applications of Equation (35) will now be covered.

Change in Fuel Requirements Associated with Decisions Available toConsumers

Consumers have four decisions available to them that affects theirenergy consumption for heating. FIG. 6 is a process flow diagramshowing, by way of example, consumer heating energy consumption-relateddecision points. These decisions 71 include:

-   -   1. Change the thermal conductivity UA^(Total) by upgrading the        building shell to be more thermally efficient (process 72).    -   2. Reduce or change the average indoor temperature by reducing        the thermostat manually, programmatically, or through a        “learning” thermostat (process 73).    -   3. Upgrade the HVAC system to increase efficiency (process 74).    -   4. Increase the solar gain by increasing the effective window        area (process 75).        Other decisions are possible. Here, these four specific options        can be evaluated supra by simply taking the derivative of        Equation (35) with respect to a variable of interest. The result        for each case is valid where UA^(Total)≥UA^(Balance Point).        Otherwise, fuel consumption is 0.

Changes associated with other internal gains, such as increasingoccupancy, increasing internal electric gains, or increasing solarheating gains, could be calculated using a similar approach.

Change in Thermal Conductivity

A change in thermal conductivity UA^(Total) can affect a change in fuelrequirements. The derivative of Equation (35) is taken with respect tothermal conductivity, which equals the average indoor minus outdoortemperatures times the number of hours divided by HVAC systemefficiency. Note that initial thermal efficiency is irrelevant in theequation. The effect of a change in thermal conductivity UA^(Total)(process 72) can be evaluated by solving:

$\begin{matrix}{\frac{{dQ}^{Fuel}}{{dUA}^{Total}} = \frac{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}{\eta^{HVAC}}} & (36)\end{matrix}$

Change in Average Indoor Temperature

A change in average indoor temperature can also affect a change in fuelrequirements. The derivative of Equation (35) is taken with respect tothe average indoor temperature. Since UA^(Balance Point) is also afunction of average indoor temperature, application of the product ruleis required. After simplifying, the effect of a change in average indoortemperature (process 73) can be evaluated by solving:

$\begin{matrix}{\frac{{dQ}^{Fuel}}{d\; \overset{\_}{T^{Indoor}}} = {( {UA}^{Total} )( \frac{H}{\eta^{HVAC}} )}} & (37)\end{matrix}$

Change in HVAC System Efficiency

As well, a change in HVAC system efficiency can affect a change in fuelrequirements. The derivative of Equation (35) is taken with respect toHVAC system efficiency, which equals current fuel consumption divided byHVAC system efficiency. Note that this term is not linear withefficiency and thus is valid for small values of efficiency changes. Theeffect of a change in fuel requirements relative to the change in HVACsystem efficiency (process 74) can be evaluated by solving:

$\begin{matrix}{\frac{d\; Q^{Fuel}}{d\; \eta^{HVAC}} = {- {Q^{Fuel}( \frac{H}{\eta^{HVAC}} )}}} & (38)\end{matrix}$

Change in Solar Gains

An increase in solar gains can be accomplished by increasing theeffective area of south-facing windows. Effective area can be increasedby trimming trees blocking windows, removing screens, cleaning windows,replacing windows with ones that have higher SHGCs, installingadditional windows, or taking similar actions. In this case, thevariable of interest is the effective window area W. The total gain persquare meter of additional effective window area equals the availableresource (kWh/m²) divided by HVAC system efficiency, converted to Btus.The derivative of Equation (35) is taken with respect to effectivewindow area. The effect of an increase in solar gains (process 74) canbe evaluated by solving:

$\begin{matrix}{\frac{d\; Q^{Fuel}}{d\; \overset{\_}{W}} = {{- \lbrack \frac{( \overset{\_}{Solar} )(H)}{\eta^{HVAC}} \rbrack}( \frac{3\text{,}412\mspace{14mu} {Btu}}{kWh} )}} & (39)\end{matrix}$

Discussion

Both Equations (1) and (35) provide ways to calculate fuel consumptionrequirements. The two equations differ in several key ways:

-   -   1. UA^(Total) only occurs in one place in Equation (35), whereas        Equation (1) has multiple indirect and non-linear dependencies        to UA^(Total).    -   2. UA^(Total) is divided into two parts in Equation (35), while        there is only one occurrence of UA^(Total) in Equation (1).    -   3. The concept of balance point thermal conductivity in        Equation (35) replaces the concept of balance point temperature        in Equation (1).    -   4. Heat from occupants, electricity consumption, and solar gains        are grouped together in Equation (35) as internal heating gains,        while these values are treated separately in Equation (1).

Second, Equations (29) through (32) provide empirical methods todetermine both the point at which a building has no auxiliary heatingrequirements and the current thermal conductivity. Equation (1)typically requires a full detailed energy audit to obtain the datarequired to derive thermal conductivity. In contrast, Equations (25)through (28), as applied through the first approach, can substantiallyreduce the scope of an energy audit.

Third, both Equation (4) and Equation (36) provide ways to calculate achange in fuel requirements relative to a change in thermalconductivity. However, these two equations differ in several key ways:

-   -   1. Equation (4) is complex, while Equation (36) is simple.    -   2. Equation (4) depends upon current building thermal        conductivity, balance point temperature, solar savings fraction,        auxiliary heating efficiency, and a variety of other        derivatives. Equation (36) only requires the auxiliary heating        efficiency in terms of building-specific information.

Equation (36) implies that, as long as some fuel is required forauxiliary heating, a reasonable assumption, a change in fuelrequirements will only depend upon average indoor temperature (asapproximated by thermostat setting), average outdoor temperature, thenumber of hours (or other time units) in the (heating) season, and HVACsystem efficiency. Consequently, any building shell (or envelope)investment can be treated as an independent investment. Importantly,Equation (36) does not require specific knowledge about buildingconstruction, age, occupancy, solar gains, internal electric gains, orthe overall thermal conductivity of the building. Only thecharacteristics of the portion of the building that is being replaced,the efficiency of the HVAC system, the indoor temperature (as reflectedby the thermostat setting), the outdoor temperature (based on location),and the length of the winter season are required; knowledge about therest of the building is not required. This simplification is a powerfuland useful result.

Fourth, Equation (37) provides an approach to assessing the impact of achange in indoor temperature, and thus the effect of making a change inthermostat setting. Note that Equation (31) only depends upon theoverall efficiency of the building, that is, the building's totalthermal conductivity UA^(Total), the length of the winter season (innumber of hours or other time units), and the HVAC system efficiency;Equation (31) does not depend upon either the indoor or outdoortemperature.

Equation (31) is useful in assessing claims that are made by HVACmanagement devices, such as the Nest thermostat device, manufactured byNest Labs, Inc., Palo Alto, Calif., or the Lyric thermostat device,manufactured by Honeywell Int'l Inc., Morristown, N.J., or otherso-called “smart” thermostat devices. The fundamental idea behind thesetypes of HVAC management devices is to learn behavioral patterns, sothat consumers can effectively lower (or raise) their average indoortemperatures in the winter (or summer) months without affecting theirpersonal comfort. Here, Equation (31) could be used to estimate thevalue of heating and cooling savings, as well as to verify the consumerbehaviors implied by the new temperature settings.

Balance Point Temperature

Before leaving this section, balance point temperature should briefly bediscussed. The formulation in this first approach does not involvebalance point temperature as an input. A balance point temperature,however, can be calculated to equal the point at which there is no fuelconsumption, such that there are no gains associated with auxiliaryheating (Q^(Gains-Auxiliary Heating) equals 0) and the auxiliary heatingthermal conductivity (UA^(Auxiliary Heating) in Equation (32)) is zero.Inserting these assumptions into Equation (20) and labeling T^(Outdoor)as T^(Balance Point) yields:

Q ^(Gains-Internal) =UA ^(Total)( T ^(Indoor) −T^(Balance Point))(H)  (40)

Equation (40) simplifies to:

$\begin{matrix}{{{\overset{\_}{T}}^{{Balance}\mspace{14mu} {Point}} = {{\overset{\_}{T}}^{Indoor} - {\frac{{\overset{\_}{Q}}^{{Gains} - {Internal}}}{{UA}^{Total}}\mspace{14mu} {where}}}}{{\overset{\_}{Q}}^{{Gains} - {Internal}} = \frac{Q^{{Gains} - {Internal}}}{H}}} & (41)\end{matrix}$

Equation (41) is identical to Equation (2), except that average valuesare used for indoor temperature T ^(Indoor), balance point temperature T^(Balance Point), and fuel consumption for internal heating gains Q^(Gains-Internal), and that heating gains from occupancy(Q^(Gains-Occupants)), electric (Q^(Gains-Electric)), and solar(Q^(Gains-Solar)) are all included as part of internal heating gains(Q^(Gains-Internal)).

Application: Change in Thermal Conductivity Associated with OneInvestment

An approach to calculating a new value for total thermal conductivity

^(Total) after a series of M changes (or investments) are made to abuilding is described in commonly-assigned U.S. patent application,entitled “System and Method for Interactively Evaluating PersonalEnergy-Related Investments,” Ser. No. 14/294,079, filed Jun. 2, 2014,pending, the disclosure of which is incorporated by reference. Theapproach is summarized therein in Equation (41), which provides:

Total = UA Total + ∑ j = 1 M  ( U j - U ^ j )  A j + ρ   c  ( n - n^ )  V ( 42 )

where a caret symbol (̂) denotes a new value, infiltration losses arebased on the density of air (φ, specific heat of air (c), number of airchanges per hour (n), and volume of air per air change (V). In addition,U^(j) and Û^(j) respectively represent the existing and proposedU-values of surface j, and A^(j) represents the surface area of surfacej. The volume of the building V can be approximated by multiplyingbuilding square footage by average ceiling height. The equation, with aslight restatement, equals:

Total = UA Total + Δ  UA Total ( 43 ) and Δ  UA Total = ∑ j = 1 M  (U j - U ^ j )  A j + ρc  ( n - n ~ )  V . ( 44 )

If there is only one investment, the m superscripts can be dropped andthe change in thermal conductivity UA^(Total) equals the area (A) timesthe difference of the inverse of the old and new R-values R and{circumflex over (R)}:

$\begin{matrix}{{\Delta {UA}}^{Total} = {{A( {U - \hat{U}} )} = {{A( {\frac{1}{R} - \frac{1}{\hat{R}}} )}.}}} & (45)\end{matrix}$

Fuel Savings

The fuel savings associated with a change in thermal conductivityUA^(Total) for a single investment equals Equation (45) times (36):

$\begin{matrix}{{\Delta Q}^{Fuel} = {{{\Delta {UA}}^{Total}\frac{{dQ}^{Fuel}}{{dUA}^{Total}}} = {{A( {\frac{1}{R} - \frac{1}{\hat{R}}} )}\frac{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}{\eta^{HVAC}}}}} & (46)\end{matrix}$

where ΔQ^(Fuel) signifies the change in fuel consumption.

Economic Value

The economic value of the fuel savings (Annual Savings) equals the fuelsavings times the average fuel price (Price) for the building inquestion:

$\begin{matrix}{{{Annual}\mspace{14mu} {Savings}} = {{A( {\frac{1}{R} - \frac{1}{\hat{R}}} )}\frac{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}{\eta^{HVAC}}({Price})}} & (47) \\{where} & \; \\{{Price} = \{ \begin{matrix}{\frac{{Price}^{NG}}{10^{5}}\mspace{14mu} {if}\mspace{14mu} {price}\mspace{14mu} {has}\mspace{14mu} {units}\mspace{14mu} {of}\mspace{14mu} \$ \mspace{14mu} {per}\mspace{14mu} {therm}} \\{\frac{{Price}^{Electrity}}{3,412}\mspace{14mu} {if}\mspace{14mu} {price}\mspace{14mu} {has}\mspace{11mu} {units}\mspace{11mu} {of}\mspace{14mu} \$ \mspace{14mu} {per}\mspace{14mu} {kWh}}\end{matrix} } & \;\end{matrix}$

where Price^(NG) represents the price of natural gas andPrice^(Electricity) represents the price of electricity. Other pricingamounts, pricing conversion factors, or pricing expressions arepossible.

Example

Consider an example. A consumer in Napa, Calif. wants to calculate theannual savings associating with replacing a 20 ft² single-pane windowthat has an R-value of 1 with a high efficiency window that has anR-value of 4. The average temperature in Napa over the 183-day winterperiod (4,392 hours) from October 1 to March 31 is 50° F. The consumersets his thermostat at 68° F., has a 60 percent efficient natural gasheating system, and pays $1 pays $1 per therm for natural gas. How muchmoney will the consumer save per year by making this change?

Putting this information into Equation (47) suggests that he will save$20 per year:

$\begin{matrix}{{{Annual}\mspace{14mu} {Savings}} = {{20( {\frac{1}{1} - \frac{1}{4}} )\frac{( {68 - 50} )( {4,392} )}{0.6}( \frac{1}{10^{5}} )} = {\$ 20}}} & (48)\end{matrix}$

Application: Validate Building Shell Improvements Savings

Many energy efficiency programs operated by power utilities grapple withthe issue of measurement and evaluation (M&E), particularly with respectto determining whether savings have occurred after building shellimprovements were made. Equations (29) through (32) can be applied tohelp address this issue. These equations can be used to calculate abuilding's total thermal conductivity UA^(Total). This result providesan empirical approach to validating the benefits of building shellinvestments using measured data.

Equations (29) through (32) require the following inputs:

1) Weather:

-   -   a) Average outdoor temperature (° F.).    -   b) Average indoor temperature (° F.).    -   c) Average direct solar resource on a vertical, south-facing        surface.

2) Fuel and energy:

-   -   a) Average gross indoor electricity consumption.    -   b) Average natural gas fuel consumption for space heating.    -   c) Average electric fuel consumption for space heating.

3) Other inputs:

-   -   a) Average number of occupants.    -   b) Effective window area.    -   c) HVAC system efficiency.

Weather data can be determined as follows. Indoor temperature can beassumed based on the setting of the thermostat (assuming that thethermostat's setting remained constant throughout the time period), ormeasured and recorded using a device that takes hourly or periodicindoor temperature measurements, such as a Nest thermostat device or aLyric thermostat device, cited supra, or other so-called “smart”thermostat devices. Outdoor temperature and solar resource data can beobtained from a service, such as Solar Anywhere SystemCheck, citedsupra, or the National Weather Service. Other sources of weather dataare possible.

Fuel and energy data can be determined as follows. Monthly utilitybilling records provide natural gas consumption and net electricitydata. Gross indoor electricity consumption can be calculated by addingPV production, whether simulated using, for instance, the Solar AnywhereSystemCheck service, cited supra, or measured directly, and subtractingout external electricity consumption, that is, electricity consumptionfor electric devices that do not deliver all heat that is generated intothe interior of the building. External electricity consumption includeselectric vehicle (EV) charging and electric water heating. Other typesof external electricity consumption are possible. Natural gasconsumption for heating purposes can be estimated by subtractingnon-space heating consumption, which can be estimated, for instance, byexamining summer time consumption using an approach described incommonly-assigned U.S. patent application, entitled “System and Methodfor Facilitating Implementation of Holistic Zero Net EnergyConsumption,” Ser. No. 14/531,933, filed Nov. 3, 2014, pending, thedisclosure of which is incorporated by reference. Other sources of fueland energy data are possible.

Finally, the other inputs can be determined as follows. The averagenumber of occupants can be estimated by the building owner or occupant.Effective window area can be estimated by multiplying actualsouth-facing window area times solar heat gain coefficient (estimated orbased on empirical tests, as further described infra), and HVAC systemefficiency can be estimated (by multiplying reported furnace ratingtimes either estimated or actual duct system efficiency), or can bebased on empirical tests, as further described infra. Other sources ofdata for the other inputs are possible.

Consider an example. FIG. 7 is a table 80 showing, by way of example,data used to calculate thermal conductivity. The data inputs are for asample house in Napa, Calif. based on the winter period of October 1 toMarch 31 for six winter seasons, plus results for a seventh winterseason after many building shell investments were made. (Note thebuilding improvements facilitated a substantial increase in the averageindoor temperature by preventing a major drop in temperature duringnight-time and non-occupied hours.) South-facing windows had aneffective area of 10 m² and the solar heat gain coefficient is estimatedto be 0.25 for an effective window area of 2.5 m². The measured HVACsystem efficiency of 59 percent was based on a reported furnaceefficiency of 80 percent and an energy audit-based duct efficiency of 74percent.

FIG. 8 is a table 90 showing, by way of example, thermal conductivityresults for each season using the data in the table 80 of FIG. 7 asinputs into Equations (29) through (32). Thermal conductivity is inunits of Btu/h-° F. FIG. 9 is a graph 100 showing, by way of example, aplot of the thermal conductivity results in the table 90 of FIG. 8. Thex-axis represents winter seasons for successive years, each winterseason running from October 1 to March 31. The y-axis represents thermalconductivity. The results from a detailed energy audit, performed inearly 2014, are superimposed on the graph. The energy audit determinedthat the house had a thermal conductivity of 773 Btu/h-° F. The averageresult estimated for the first six seasons was 791 Btu/h-° F. A majoramount of building shell work was performed after the 2013-2014 winterseason, and the results show a 50-percent reduction in heating energyconsumption in the 2014-2015 winter season.

Application: Evaluate Investment Alternatives

The results of this work can be used to evaluate potential investmentalternatives. FIG. 10 is a graph 110 showing, by way of example, anauxiliary heating energy analysis and energy consumption investmentoptions. The x-axis represents total thermal conductivity, UA^(Total) inunits of Btu/hr-° F. The y-axis represents total heating energy. Thegraph presents the analysis of the Napa, Calif. building from theearlier example, supra, using the equations previously discussed. Thethree lowest horizontal bands correspond to the heat provided throughinternal gains 111, including occupants, heat produced by operatingelectric devices, and solar heating. The solid circle 112 represents theinitial situation with respect to heating energy consumption. Thediagonal lines 113 a, 113 b, 113 c represent three alternative heatingsystem efficiencies versus thermal conductivity (shown in the graph asbuilding losses). The horizontal dashed line 114 represents an option toimprove the building shell and the vertical dashed line 115 representsan option to switch to electric resistance heating. The plain circle 116represents the final situation with respect to heating energyconsumption.

Other energy consumption investment options (not depicted) are possible.These options include switching to an electric heat pump, increasingsolar gain through window replacement or tree trimming (this optionwould increase the height of the area in the graph labeled “SolarGains”), or lowering the thermostat setting. These options can becompared using the approach described with reference to Equations (25)through (28) to compare the options in terms of their costs and savings,which will help the homeowner to make a wiser investment.

Second Approach: Time Series Fuel Consumption

The previous section presented an annual fuel consumption model. Thissection presents a detailed time series model. This section alsocompares results from the two methods and provides an example of how toapply the on-site empirical tests.

Building-Specific Parameters

The building temperature model used in this second approach requiresthree building parameters: (1) thermal mass; (2) thermal conductivity;and (3) effective window area. FIG. 11 is a functional block diagramshowing thermal mass, thermal conductivity, and effective window arearelative to a structure 121. By way of introduction, these parameterswill now be discussed.

Thermal Mass (M)

The heat capacity of an object equals the ratio of the amount of heatenergy transferred to the object and the resulting change in theobject's temperature. Heat capacity is also known as “thermalcapacitance” or “thermal mass” (122) when used in reference to abuilding. Thermal mass Q is a property of the mass of a building thatenables the building to store heat, thereby providing “inertia” againsttemperature fluctuations. A building gains thermal mass through the useof building materials with high specific heat capacity and high density,such as concrete, brick, and stone.

The heat capacity is assumed to be constant when the temperature rangeis sufficiently small. Mathematically, this relationship can beexpressed as:

Q _(Δt) =M(T _(t+Δt) ^(Indoor) −T _(t) ^(Indoor))  (49)

where M equals the thermal mass of the building and temperature units Tare in ° F. Q is typically expressed in Btu or Joules. In that case, Mhas units of Btu/° F. Q can also be divided by 1 kWh/3,412 Btu toconvert to units of kWh/° F.

Thermal Conductivity (UA^(Total))

The building's thermal conductivity UA^(Total) (123) is the amount ofheat that the building gains or losses as a result of conduction andinfiltration. Thermal conductivity UA^(Total) was discussed supra withreference to the first approach for modeling annual heating fuelconsumption.

Effective Window Area (W)

The effective window area (in units of m²) (124), also discussed indetail supra, specifies how much of an available solar resource isabsorbed by the building. Effective window area is the dominant means ofsolar gain in a typical building during the winter and includes theeffect of physical shading, window orientation, and the window's solarheat gain coefficient. In the northern hemisphere, the effective windowarea is multiplied by the available average direct irradiance on avertical, south-facing surface (kW/m²), times the amount of time (H) toresult in the kWh obtained from the windows.

Energy Gain or Loss

The amount of heat transferred to or extracted from a building (Q) overa time period of Δt is based on a number of factors, including:

-   -   1) Loss (or gain if outdoor temperature exceeds indoor        temperature) due to conduction and infiltration and the        differential between the indoor and outdoor temperatures.    -   2) Gain, when the HVAC system is in the heating mode, or loss,        when the HVAC system is in the cooling mode.    -   3) Gain associated with:        -   a) Occupancy and heat given off by people.        -   b) Heat produced by consuming electricity inside the            building.        -   c) Solar radiation.

Mathematically, Q can be expressed as:

$\begin{matrix}{Q_{\Delta t} = {\quad{\lbrack {\overset{\overset{{Envelope}\mspace{14mu} {Gain}\mspace{14mu} {or}\mspace{14mu} {Loss}}{}}{{UA}^{Total}( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} )} + \overset{\overset{{Occupancy}\mspace{14mu} {Gain}}{}}{(250)\overset{\_}{P}} + \overset{\overset{{Internal}\mspace{14mu} {Electric}\mspace{14mu} {Gain}}{}}{\overset{\_}{Electric}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )} + \overset{\overset{{Solar}\mspace{14mu} {Gain}}{}}{W{\overset{\_}{Solar}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )}} + \overset{\overset{{HVAC}\mspace{14mu} {Gain}\mspace{14mu} {or}\mspace{14mu} {Loss}}{}}{({HeatOrCool})R^{HVAC}\eta^{HVAC}\overset{\_}{Status}}} \rbrack {\Delta t}}}} & (50)\end{matrix}$

where:

-   -   Except as noted otherwise, the bars over the variable names        represent the average value over Δt hours, that is, the duration        of the applicable empirical test. For instance, T ^(Outdoor)        represents the average outdoor temperature between the time        interval of t and t+Δt.    -   UA^(Total) is the thermal conductivity (in units of Btu/hour-°        F.).    -   W is the effective window area (in units of m²).    -   Occupancy Gain is based on the average number of people (P) in        the building during the applicable empirical test (and the heat        produced by those people). The average person is assumed to        produce 250 Btu/hour.    -   Internal Electric Gain is based on heat produced by indoor        electricity consumption (Electric), as averaged over the        applicable empirical test, but excludes electricity for purposes        that do not produce heat inside the building, for instance,        electric hot water heating where the hot water is discarded down        the drain, or where there is no heat produced inside the        building, such as is the case with EV charging.    -   Solar Gain is based on the average available normalized solar        irradiance (Solar) during the applicable empirical test (with        units of kW/m²). This value is the irradiance on a vertical        surface to estimate solar received on windows; global horizontal        irradiance (GHI) can be used as a proxy for this number when W        is allowed to change on a monthly basis.    -   HVAC Gain or Loss is based on whether the HVAC is in heating or        cooling mode (GainOrLoss is 1 for heating and −1 for cooling),        the rating of the HVAC system (R in Btu), HVAC system efficiency        (η^(HVAC), including both conversion and delivery system        efficiency), average operation status (Status) during the        empirical test, a time series value that is either off (0        percent) or on (100 percent),    -   Other conversion factors or expressions are possible.

Energy Balance

Equation (49) reflects the change in energy over a time period andequals the product of the temperature change and the building's thermalmass. Equation (50) reflects the net gain in energy over a time periodassociated with the various component sources. Equation (49) can be setto equal Equation (50), since the results of both equations equal thesame quantity and have the same units (Btu). Thus, the total heat changeof a building will equal the sum of the individual heat gain/losscomponents:

$\begin{matrix}{\overset{\overset{{Total}\mspace{14mu} {Heat}\mspace{14mu} {Change}}{}}{M( {T_{t + {\Delta t}}^{Indoor} - T_{t}^{Indoor}} )} = {\quad{\lbrack {\overset{\overset{{Envelope}\mspace{14mu} {Gain}\mspace{14mu} {or}\mspace{14mu} {Loss}}{}}{{UA}^{Total}( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} )} + \overset{\overset{{Occupancy}\mspace{14mu} {Gain}}{}}{(250)\overset{\_}{P}} + \overset{\overset{{Internal}\mspace{14mu} {Electric}\mspace{14mu} {Gain}}{}}{\overset{\_}{Electric}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )} + \overset{\overset{{Solar}\mspace{14mu} {Gain}}{}}{W{\overset{\_}{Solar}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )}} + \overset{\overset{{HVAC}\mspace{14mu} {Gain}\mspace{14mu} {or}\mspace{14mu} {Loss}}{}}{({HeatOrCool})R^{HVAC}\eta^{HVAC}\overset{\_}{Status}}} \rbrack {\Delta t}}}} & (51)\end{matrix}$

Equation (51) can be used for several purposes. FIG. 12 is a flowdiagram showing a computer-implemented method 130 for modeling intervalbuilding heating energy consumption in accordance with a furtherembodiment. Execution of the software can be performed with theassistance of a computer system, such as further described infra withreference to FIG. 29, as a series of process or method modules or steps.

As a single equation, Equation (51) is potentially very useful, despitehaving five unknown parameters. In this second approach, the unknownparameters are solved by performing a series of short duration empiricaltests (step 131), as further described infra with reference to FIG. 14.Once the values of the unknown parameters are found, a time series ofindoor temperature data can be constructed (step 132), which will thenallow annual fuel consumption to be calculated (step 133) and maximumindoor temperature to be found (step 134). The short duration tests willfirst be discussed.

Empirically Determining Building- and Equipment-Specific ParametersUsing Short Duration Tests

A series of tests can be used to iteratively solve Equation (51) toobtain the values of the unknown parameters by ensuring that theportions of Equation (51) with the unknown parameters are equal to zero.These tests are assumed to be performed when the HVAC is in heating modefor purposes of illustration. Other assumptions are possible.

FIG. 13 is a table 140 showing the characteristics of empirical testsused to solve for the five unknown parameters in Equation (51). Theempirical test characteristics are used in a series ofsequentially-performed short duration tests; each test builds on thefindings of earlier tests to replace unknown parameters with foundvalues.

The empirical tests require the use of several components, including acontrol for turning an HVAC system ON or OFF, depending upon the test;an electric controllable interior heat source; a monitor to measure theindoor temperature during the test; a monitor to measure the outdoortemperature during the test; and a computer or other computationaldevice to assemble the test results and finding thermal conductivity,thermal mass, effective window area, and HVAC system efficiency of abuilding based on the findings. The components can be separate units, orcould be consolidated within one or more combined units. For instance, acomputer equipped with temperature probes could both monitor, record andevaluate temperature findings. FIG. 14 is a flow diagram showing aroutine 150 for empirically determining building- and equipment-specificparameters using short duration tests for use in the method 130 of FIG.12. The approach is to run a serialized series of empirical tests. Thefirst test solves for the building's total thermal conductivity(UA^(Total)) (step 151). The second test uses the empirically-derivedvalue for UA^(Total) to solve for the building's thermal mass (M) (step152). The third test uses both of these results, thermal conductivityand thermal mass, to find the building's effective window area (W) (step153). Finally, the fourth test uses the previous three test results todetermine the overall HVAC system efficiency (step 145). Consider how toperform each of these tests.

Test 1: Building Thermal Conductivity (UA^(Total))

The first step is to find the building's total thermal conductivity(UA^(Total)) (step 151). Referring back to the table in FIG. 13, thisshort-duration test occurs at night (to avoid any solar gain) with theHVAC system off (to avoid any gain from the HVAC system), and by havingthe indoor temperature the same at the beginning and the ending of thetest by operating an electric controllable interior heat source, such asportable electric space heaters that operate at 100% efficiency, so thatthere is no change in the building temperature's at the beginning and atthe ending of the test. Thus, the interior heart source must havesufficient heating capacity to maintain the building's temperaturestate. Ideally, the indoor temperature would also remain constant toavoid any potential concerns with thermal time lags.

These assumptions are input into Equation (51):

$\begin{matrix}{{M(0)} = {\lbrack {{{UA}^{Total}( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} )} + {(250)\overset{\_}{P}} + {\overset{\_}{Electric}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )} + {{W(0)}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )} + {(1)R^{HVAC}{\eta^{HVAC}(0)}}} \rbrack {\Delta t}}} & (52)\end{matrix}$

The portions of Equation (52) that contain four of the five unknownparameters now reduce to zero. The result can be solved for UA^(Total):

$\begin{matrix}{{UA}^{Total} = \frac{\lbrack {{(250)\overset{\_}{P}} + {\overset{\_}{Electric}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )}} \rbrack}{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )}} & (53)\end{matrix}$

where T ^(Indoor) represents the average indoor temperature during theempirical test, T ^(Outdoor) represents the average outdoor temperatureduring the empirical test, P represents the average number of occupantsduring the empirical test, and Electric represents average indoorelectricity consumption during the empirical test.

Equation (53) implies that the building's thermal conductivity can bedetermined from this test based on average number of occupants, averagepower consumption, average indoor temperature, and average outdoortemperature.

Test 2: Building Thermal Mass (M)

The second step is to find the building's thermal mass (M) (step 152).This step is accomplished by constructing a test that guarantees M isspecifically non-zero since UA^(Total) is known based on the results ofthe first test. This second test is also run at night, so that there isno solar gain, which also guarantees that the starting and the endingindoor temperatures are not the same, that is, T_(t+Δt) ^(Indoor)≠T_(t)^(Indoor), respectively at the outset and conclusion of the test by notoperating the HVAC system. These assumptions are input into Equation(51) and solving yields a solution for M:

$\begin{matrix}{M = {\lbrack \frac{\begin{matrix}{{{UA}^{Total}( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} )} + {(250)\overset{\_}{P}} +} \\{\overset{\_}{Electric}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )}\end{matrix}}{( {T_{t + {\Delta t}}^{Indoor} - T_{t}^{Outdoor}} )} \rbrack {\Delta t}}} & (54)\end{matrix}$

where UA^(Total) represents the thermal conductivity, T ^(Indoor)represents the average indoor temperature during the empirical test, T^(Outdoor) represents the average outdoor temperature during theempirical test, P represents the average number of occupants during theempirical test, Electric represents average indoor electricityconsumption during the empirical test, t represents the time at thebeginning of the empirical test, Δt represents the duration of theempirical test, T_(t+Δt) ^(Indoor) represents the ending indoortemperature, T_(t) ^(Indoor) represents the starting indoor temperature,and T_(t+Δt) ^(Indoor)≠T_(t) ^(Indoor).

Test 3: Building Effective Window Area (W)

The third step to find the building's effective window area (W) (step153) requires constructing a test that guarantees that solar gain isnon-zero. This test is performed during the day with the HVAC systemturned off. Solving for W yields:

$\begin{matrix}{W = {\{ {\lbrack \frac{M( {T_{t + {\Delta t}}^{Indoor} - T_{t}^{Indoor}} )}{3,412{\Delta t}} \rbrack - \frac{{UA}^{Total}( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} )}{3,412} - \frac{(250)\overset{\_}{P}}{3,412} - \overset{\_}{Electric}} \} \lbrack \frac{1}{\overset{\_}{Solar}} \rbrack}} & (55)\end{matrix}$

where M represents the thermal mass, t represents the time at thebeginning of the empirical test, Δt represents the duration of theempirical test, T_(t+Δt) ^(Indoor) represents the ending indoortemperature, and T^(Indoor) represents the starting indoor temperature,UA^(Total) represents the thermal conductivity, T ^(Indoor) representsthe average indoor temperature, T ^(Outdoor) represents the averageoutdoor temperature, P represents the average number of occupants duringthe empirical test, Electric represents average electricity consumptionduring the empirical test, and Solar represents the average solar energyproduced during the empirical test.

Test 4: HVAC System Efficiency (η^(Furnace)η^(Delivery))

The fourth step determines the HVAC system efficiency (step 154). TotalHVAC system efficiency is the product of the furnace efficiency and theefficiency of the delivery system, that is, the duct work and heatdistribution system. While these two terms are often solved separately,the product of the two terms is most relevant to building temperaturemodeling. This test is best performed at night, so as to eliminate solargain. Thus:

$\begin{matrix}{ {\eta^{HVAC} = {\lbrack \frac{M( {T_{t + {\Delta t}}^{Indoor} - T_{t}^{Indoor}} )}{\Delta t} \rbrack - {{UA}^{Total}( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} )} - {(250)\overset{\_}{P}} - \overset{\_}{Electric}}} \} {( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )\lbrack \frac{1}{(1)R^{HVAC}\overset{\_}{Status}} \rbrack}} & (56)\end{matrix}$

where M represents the thermal mass, t represents the time at thebeginning of the empirical test, Δt represents the duration of theempirical test, T_(t+Δt) ^(Indoor) represents the ending indoortemperature, and T_(t) ^(Indoor) represents the starting indoortemperature, UA^(Total) represents the thermal conductivity, T ^(Indoor)represents the average indoor temperature, T ^(Outdoor) represents theaverage outdoor temperature, P represents the average number ofoccupants during the empirical test, Electric represents averageelectricity consumption during the empirical test, Status represents theaverage furnace operation status, and R^(Furnace) represents the ratingof the furnace.

Note that HVAC duct efficiency can be determined without performing aduct leakage test if the generation efficiency of the furnace is known.This observation usefully provides an empirical method to measure ductefficiency without having to perform a duct leakage test.

Time Series Indoor Temperature Data

The previous subsection described how to perform a series of empiricalshort duration tests to determine the unknown parameters in Equation(51). Commonly-assigned U.S. patent application Ser. No. 14/531,933,cited supra, describes how a building's UA^(Total) can be combined withhistorical fuel consumption data to estimate the benefit of improvementsto a building. While useful, estimating the benefit requires measuredtime series fuel consumption and HVAC system efficiency data. Equation(51), though, can be used to perform the same analysis without the needfor historical fuel consumption data.

Referring back to FIG. 12, Equation (51) can be used to construct timeseries indoor temperature data (step 132) by making an approximation.Let the time period (Δt) be short (an hour or less), so that the averagevalues are approximately equal to the value at the beginning of the timeperiod, that is, assume T^(Outdoor) ≈T_(t) ^(Outdoor). The averagevalues in Equation (51) can be replaced with time-specific subscriptedvalues and solved to yield the final indoor temperature.

$\begin{matrix}{T_{t + {\Delta t}}^{Indoor} = {T_{t}^{Indoor} + {{\lbrack \frac{1}{M} \rbrack\lbrack {{{UA}^{Total}( {T_{t}^{Outdoor} - T_{t}^{Indoor}} )} + {(250)P_{t}} + {{Electric}_{t}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )} + {{WSolar}_{t}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )} + {({HeatOrCool})R^{HVAC}\eta^{HVAC}{Status}_{t}}} \rbrack}{\Delta t}}}} & (57)\end{matrix}$

Once T_(t+Δt) ^(Indoor) is known, Equation (57) can be used to solve forT_(t+2Δt) ^(Indoor) and so on.

Importantly, Equation (57) can be used to iteratively construct indoorbuilding temperature time series data with no specific information aboutthe building's construction, age, configuration, number of stories, andso forth. Equation (57) only requires general weather datasets (outdoortemperature and irradiance) and building-specific parameters. Thecontrol variable in Equation (57) is the fuel required to deliver theauxiliary heat at time t, as represented in the Status variable, thatis, at each time increment, a decision is made whether to run the HVACsystem.

Seasonal Fuel Consumption

Equation (51) can also be used to calculate seasonal fuel consumption(step 133) by letting Δt equal the number of hours (H) in the entireseason, either heating or cooling (and not the duration of theapplicable empirical test), rather than making Δt very short (such as anhour, as used in an applicable empirical test). The indoor temperatureat the start and the end of the season can be assumed to be the same or,alternatively, the total heat change term on the left side of theequation can be assumed to be very small and set equal to zero.Rearranging Equation (51) provides:

$\begin{matrix}{{({HeatOrCool})R^{HVAC}\eta^{HVAC}{\overset{\_}{Status}(H)}} = {{{- \lbrack {{UA}^{Total}( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} )} \rbrack}(H)} - {\lbrack {{(250)P} + {\overset{\_}{Electric}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )} + {W{\overset{\_}{Solar}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )}}} \rbrack (H)}}} & (58)\end{matrix}$

Total seasonal fuel consumption based on Equation (51) can be shown tobe identical to fuel consumption calculated using the annual methodbased on Equation (35). First, Equation (58), which is a rearrangementof Equation (51), can be simplified. Multiplying Equation (58) byHeatOrCool results in (HeatOrCool)² on the left hand side, which equals1 for both heating and cooling seasons, and can thus be dropped from theequation. In addition, the sign on the first term on the right hand sideof Equation (58) ([UA^(Total)(T ^(Outdoor)+T ^(Indoor))](H)) can bechanged by reversing the order of the temperatures. Per Equation (9),the second term on the right hand side of the equation

$( {\lbrack {{(250)\overset{\_}{P}} + {\overset{\_}{Electric}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )} + {W\; {\overset{\_}{Solar}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )}}} \rbrack (H)} )$

equals internal gains (Q^(Gains-Internal)), which can be substitutedinto Equation (58). Finally, dividing the equation by HVAC efficiencyη^(HVAC) yields:

$\begin{matrix}{{R^{HVAC}{\overset{\_}{Status}(H)}} = {\lbrack {{({HeatOrCool})( {UA}^{Total} )( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)} - {({HeatOrCool})Q^{{Gains} - {internal}}}} \rbrack ( \frac{1}{\eta^{HVAC}} )}} & (59)\end{matrix}$

Equation (59), which is a simplification of Equation (58), can be usedto calculate net savings in fuel, cost, and carbon emissions(environmental), as described, for instance, in commonly-assigned U.S.patent application, entitled “System and Method for Estimating IndoorTemperature Time Series Data of a Building with the Aid of a DigitalComputer,” Ser. No. 15/096,185, filed Apr. 11, 2016, pending, thedisclosure of which is incorporated by reference. Next, substitutingEquation (24) into Equation (59):

$\begin{matrix}{{R^{HVAC}{\overset{\_}{Status}(H)}} = {\quad{\quad\lbrack {({HeatOrCool}) ( {UA}^{Total} ) ( {{\overset{\_}{T}}^{Indoor} - { \quad{\overset{\_}{T}}^{Outdoor} )(H)} - {\quad{\quad{({HeatOrCool}) ({HeatOrCool}) ( {UA}^{{Balance}\mspace{14mu} {Point}} ) ( {{\overset{\_}{T}}^{Indoor} - { \quad{\overset{\_}{T}}^{Outdoor} )(H)}} \rbrack ( \frac{1}{\eta^{HVAC}} )}}}} } }}} & (60)\end{matrix}$

Once again, HeatOrCool² equals 1 for both heating and cooling seasonsand thus is dropped. Equation (60) simplifies as:

$\begin{matrix}{{R^{HVAC}{\overset{\_}{Status}(H)}} = {\quad\frac{\begin{matrix}\lbrack {{{HeatOrCool}( {UA}^{Total} )} - ( {UA}^{{Balance}\mspace{14mu} {Point}} )} \rbrack \\{( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}\end{matrix}}{\eta^{HVAC}}}} & (61)\end{matrix}$

Consider the heating season when HeatOrCool equals 1. Equation (61)simplifies as follows.

$\begin{matrix}{Q^{Fuel} = \frac{( {{UA}^{Total} - {UA}^{{Balance}\mspace{14mu} {Point}}} )( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} )(H)}{\eta^{HVAC}}} & (62)\end{matrix}$

Equation (62) illustrates total seasonal fuel consumption based onEquation (51) is identical to fuel consumption calculated using theannual method based on Equation (35).

Consider the cooling season when HeatOrCool equals −1. Multiply Equation(62) by the first part of the right hand side by −1 and reverse thetemperatures, substitute −1 for HeatOrCool, and simplify:

$\begin{matrix}{Q^{Fuel} = \frac{( {{UA}^{Total} + {UA}^{{Balance}\mspace{14mu} {Point}}} )( {{\overset{\_}{T}}^{Outdoor} - {\overset{\_}{T}}^{Indoor}} )(H)}{\eta^{HVAC}}} & (63)\end{matrix}$

A comparison of Equations (62) and (63) shows that a leverage effectoccurs that depends upon whether the season is for heating or cooling.Fuel requirements are decreased in the heating season because internalgains cover a portion of building losses (Equation (62)). Fuelrequirements are increased in the cooling season because cooling needsto be provided for both the building's temperature gains and theinternal gains (Equation (63)).

Maximum Indoor Temperature

Allowing consumers to limit the maximum indoor temperature to some valuecan be useful from a personal physical comfort perspective. The limit ofmaximum indoor temperature (step 134) can be obtained by taking theminimum of T_(t+Δt) ^(Indoor) and T^(Indoor-Max), the maximum indoortemperature recorded for the building during the heating season. Therecan be some divergence between the annual and detailed time seriesmethods when the thermal mass of the building is unable to absorb excessheat, which can then be used at a later time. Equation (57) becomesEquation (64) when the minimum is applied.

$\begin{matrix}{T_{t + {\Delta \; t}}^{Indoor} = {{Min}\{ {T^{{Indoor} - {Max}},{T_{t}^{Indoor} + {{\lbrack \frac{1}{M} \rbrack \lbrack {{{UA}^{Total}( {T_{t}^{Outdoor} - T_{t}^{Indoor}} )} + {(250)P_{t}} + {{Electric}_{t}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )} + {{WSolar}_{t}( \frac{3,412\mspace{14mu} {Btu}}{1\mspace{14mu} {kWh}} )} + {({HeatOrCool})R^{HVAC}\eta^{HVAC}{Status}_{t}}} \rbrack}\Delta \; t}}} \}}} & (64)\end{matrix}$

Comparison to Annual Method (First Approach)

Two different approaches to calculating annual fuel consumption aredescribed herein. The first approach, per Equation (35), is asingle-line equation that requires six inputs. The second approach, perEquation (64), constructs a time series dataset of indoor temperatureand HVAC system status. The second approach considers all of theparameters that are indirectly incorporated into the first approach. Thesecond approach also includes the building's thermal mass and thespecified maximum indoor temperature, and requires hourly time seriesdata for the following variables: outdoor temperature, solar resource,internal electricity consumption, and occupancy.

Both approaches were applied to the exemplary case, discussed supra, forthe sample house in Napa, Calif. Thermal mass was 13,648 Btu/° F. andthe maximum temperature was set at 72° F. The auxiliary heating energyrequirements predicted by the two approaches was then compared. FIG. 15is a graph 160 showing, by way of example, a comparison of auxiliaryheating energy requirements determined by the hourly approach versus theannual approach. The x-axis represents total thermal conductivity,UA^(Total) in units of Btu/hr-° F. They-axis represents total heatingenergy. FIG. 15 uses the same format as the graph in FIG. 10 by applyinga range of scenarios. The red line in the graph corresponds to theresults of the hourly method. The dashed black line in the graphcorresponds to the annual method. The graph suggests that results areessentially identical, except when the building losses are very low andsome of the internal gains are lost due to house overheating, which isprevented in the hourly method, but not in the annual method.

The analysis was repeated using a range of scenarios with similarresults. FIG. 16 is a graph 170 showing, by way of example, a comparisonof auxiliary heating energy requirements with the allowable indoortemperature limited to 2° F. above desired temperature of 68° F. Here,the only cases that found any meaningful divergence occurred when themaximum house temperature was very close to the desired indoortemperature. FIG. 17 is a graph 180 showing, by way of example, acomparison of auxiliary heating energy requirements with the size ofeffective window area tripled from 2.5 m² to 7.5 m². Here, internalgains were large by tripling solar gains and there was insufficientthermal mass to provide storage capacity to retain the gains.

The conclusion is that both approaches yield essentially identicalresults, except for cases when the house has inadequate thermal mass toretain internal gains (occupancy, electric, and solar).

Example

The performance of the tests described supra using measured data can beillustrated through an example. These tests were performed between 9 PMon Jan. 29, 2015 to 6 AM on Jan. 31, 2015 on a 35 year-old, 3,000 ft²house in Napa, Calif. This time period was selected to show that all ofthe tests could be performed in less than a day-and-a-half. In addition,the difference between indoor and outdoor temperatures was not extreme,making for a more challenging situation to accurately perform the tests.

FIG. 18 is a table 190 showing, by way of example, test data. The subcolumns listed under “Data” present measured hourly indoor and outdoortemperatures, direct irradiance on a vertical south-facing surface(VDI), electricity consumption that resulted in indoor heat, and averageoccupancy. Electric space heaters were used to heat the house and theHVAC system was not operated. The first three short-duration tests,described supra, were applied to this data. The specific data used arehighlighted in gray. FIG. 19 is a table 200 showing, by way of example,the statistics performed on the data in the table 190 of FIG. 18required to calculate the three test parameters. UA^(Total) wascalculated using the data in the table of FIG. 10 and Equation (53).Thermal Mass (M) was calculated using UA^(Total), the data in the tableof FIG. 10, and Equation (54). Effective Window Area (W) was calculatedusing UA^(Total), M, the data in the table of FIG. 10, and Equation(55).

These test parameters, plus a furnace rating of 100,000 Btu/hour andassumed efficiency of 56%, can be used to generate the end-of-periodindoor temperature by substituting them into Equation (57) to yield:

$\begin{matrix}{T_{t + {\Delta \; t}}^{Indoor} = {T_{t}^{Indoor} + {{\lbrack \frac{1}{18,084} \rbrack \lbrack {{429( {T_{t}^{Outdoor} - T_{t}^{Indoor}} )} + {(250)P_{t}} + {3412\mspace{14mu} {Electric}_{t}} + {11,600\; {Solar}_{t}} + {(1)( {100,000} )(0.56){Status}_{t}}} \rbrack}\Delta \; t}}} & (65)\end{matrix}$

Indoor temperatures were simulated using Equation (65) and the requiredmeasured time series input datasets. Indoor temperature was measuredfrom Dec. 15, 2014 to Jan. 31, 2015 for the test location in Napa,Calif. The temperatures were measured every minute on the first andsecond floors of the middle of the house and averaged. FIG. 20 is agraph 210 showing, by way of example, hourly indoor (measured andsimulated) and outdoor (measured) temperatures. FIG. 21 is a graph 220showing, by way of example, simulated versus measured hourly temperaturedelta (indoor minus outdoor). FIG. 20 and FIG. 21 suggest that thecalibrated model is a good representation of actual temperatures.

Empirically Determining Infiltration Using a CO₂ Monitoring Device

As explained supra with reference to Equation (5), a building's totalthermal conductivity UA^(Total) equals heat loss or gain due toconduction plus infiltration. Being able to specifically determineinfiltration is extremely empowering. The knowledge of the degree towhich infiltration and conduction individually contribute to oradversely affect a home's total thermal conductivity enables consumersto improve total building envelope efficiency by selecting energyefficiency investments according to their specific needs to respectivelyachieve better sealing or insulation.

By way of background, the rate of increase of a tracer gas in a closedspace equals the rate of addition of the tracer gas minus the rate ofremoval of the tracer gas. This relationship can be expressed in adifferential equation that can then be solved for the concentration ofthe tracer gas at a given point in time. With slight modifications tothe basic equation, the CO₂ concentration in parts per million (ppm) attime t can be expressed as:

$\begin{matrix}{C_{t}^{Inside} = {{\lbrack \frac{(P)(S)( 10^{6} )}{nV} \rbrack ( {1 - e^{- {nt}}} )} + C^{Outside} + {( {C_{0}^{Inside} - C^{Outside}} )e^{- {nt}}}}} & (66)\end{matrix}$

where C₀ ^(Inside) is the concentration of the gas (ppm) inside thebuilding at the beginning of a test; C_(t) ^(Inside) is theconcentration of the gas (ppm) inside the building at time t; C₀^(Inside) is the concentration of the gas (ppm) inside the building atthe beginning of a test; C^(Outside) is the concentration of the gas(ppm) outside the building and does not have a time subscript becauseC^(Outside) is assumed to be constant over time; P is the number ofpeople in the building; S is the rate of addition of the CO₂ source gasin ft³ per person per hour; V is the volume of the building in ft³; n isthe number of ACH; and t is time expressed in hours. The rate ofaddition of the CO₂ source gas that a person emits (S) when resting orhaving a low activity amount of work is estimated at 0.71 ft³ (i.e.,0.020 m³) of CO₂ per hour. See, e.g.,http://www.engineeringtoolbox.com/co2-persons-d_691.html, which isincorporated by reference.

In a further embodiment, infiltration can be empirically determined bymeasuring the number of ACH in a building through the use of a devicethat monitors CO₂ concentrations, such as a Netatmo Weather Station,manufactured by Netatmo, Boulogne-Billancourt, France. As the CO₂concentration is monitored and measured, the device is interfaced withan application running on a remote mobile device, such as a smartphone,or other remotely interfaceable computational device, including apersonal computer, notebook, or tablet computer, that enables themonitored CO₂ concentration and any other measured data, such astemperature, humidity, air quality, and sound level, to be regularlytracked, retrieved and recorded, displayed, and analyzed. Based on thechange in CO₂ concentration over time, the infiltration component oftotal thermal conductivity can be determined; the conduction componentof thermal conductivity can then be found by subtracting the CO₂concentration measurement-based infiltration component from thebuilding's total thermal conductivity UA^(Total).

Ordinarily, the concentration of CO₂ inside of a building will changeover time based on several factors, including:

-   -   Initial CO₂ concentration inside the building.    -   CO₂ concentration outside the building.    -   Amount of CO₂ added to the air inside the building as a result        of occupants (or pets) breathing or, infrequently, other sources        of CO₂.    -   The infiltration rate of the outside air entering the building.        Other factors causing the CO₂ concentration inside a building to        change are possible. For instance, plants absorb CO₂, yet their        impact on indoor CO₂ concentration is generally de minimus,        unless there is a substantial number of indoor plants or the        plants are of atypically large size, like mature trees. However,        in the ordinary situation, an average number of indoor house        plants will not significantly change CO₂ concentration more than        the foregoing factors.

Empty Building Test

Infiltration, and therefore conduction, can be directly determined byempirically measuring CO₂ concentration in an empty building. FIG. 22 isa flow diagram showing a method 230 for determining infiltration of abuilding through empirical testing using a CO₂ concentration monitoringdevice, in accordance with a further embodiment. This test, referred toas the Empty Building Test, requires the use of a CO₂ concentrationmonitoring device that takes CO₂ concentration readings inside the emptybuilding over time. The CO₂ concentration readings are electronicallyrecorded, after which infiltration (based on ACH) and conduction (basedon total thermal conductivity) can be determined using, for instance, acomputing or mobile device, such as further described infra withreference to FIG. 29.

Prior to testing, a baseline of outdoor CO₂ concentration should bechosen (step 231). There are at least two options for manually makingthis determination. First, if only one CO₂ concentration monitoringdevice is available, the device can be temporarily placed outside thebuilding for at least several hours to calibrate the device and measureoutdoor CO₂ concentration before installing the device indoors. Second,if available, separate CO₂ concentration monitoring devices can beemployed for indoor and outdoor uses; the outdoor device can be used tomeasure the outdoor CO₂ concentration. Note that the two devices willneed to be checked against each other prior to deployment to ensure thatthey produce the same results under the same conditions. In lieu ofactually taking a measurement, the outdoor CO₂ concentration can simplybe assumed to be 400 ppm. Still other ways or devices to determine theoutdoor CO₂ concentration are possible.

Next, a CO₂ concentration monitoring device is installed inside thebuilding (step 232) and the initial indoor CO₂ concentration isdetermined and recorded by the device (step 233). Note that the indoorCO₂ concentration should be approximately equal to the outdoor CO₂concentration. If the outdoor and indoor CO₂ concentrations vary by anappreciable margin, the source of the disparity should first beidentified and, if possible, removed from inside the building, so thatparity of indoor and outdoor CO₂ concentrations is achieved.

The concentration of indoor CO₂ is then increased from the initialmeasured indoor CO₂ concentration (step 234). There are at least threeoptions for increasing the indoor CO₂ concentration. First, people couldbe brought in to occupy the building, so as to naturally increase theCO₂ concentration through their breathing. Second, the CO₂ concentrationcan be manually increased by obtaining dry ice (widely available atnominal cost); the dry ice is converted from solid to gaseous form byplacing the dry ice in water. For instance, one pound of dry ice canproduce 250 liters (0.25 m³) of CO₂, which is equivalent to a dozenpeople breathing for one hour since each person produces 0.02 m³ of CO₂per hour at a nominal activity level. Last, CO₂ concentration can bemanually increased by discharging a CO₂ fire extinguisher inside thebuilding; the volume of CO₂ thus released would roughly correlate to thedischarge capacity of the fire extinguisher. The cost of increasing CO₂concentration in this manner would be the cost of recharging orreplacing the CO₂ fire extinguisher.

Once the CO₂ concentration has been increased from the initial CO₂concentration as measured, the sources causing any increase of theindoor CO₂ concentration are negated inside the building and monitoringbegins (step 235). The test does not need to begin precisely when thesources are negated, so long as the test is started at some pointthereafter. Note that, in this sense, “negated” simply means removingfrom the inside of the building any sources of indoor CO₂ concentrationincrease, which may be vacating the people brought in to occupy thebuilding or eliminating sources of manual CO₂ concentration increase,such as dry ice or a CO₂ fire extinguisher.

The CO₂ concentration is then monitored and recorded over time, startingonce the building has been vacated (step 236). The test can last aslittle as one hour. The monitoring and recording of the CO₂concentration is stopped when the CO₂ concentration has stabilized (step237), which occurs when the indoor CO₂ concentration is roughly equal tothe outdoor CO₂ concentration as measured prior to testing, assumingthat the indoor CO₂ concentration as initially measured wasapproximately equal to the outdoor CO₂ concentration or, alternatively,no further changes in CO₂ concentration are observed after a long lapseof time.

Based on the recorded CO₂ concentration measurements, infiltration (andtherefore conduction) can be determined (step 238), as follows. First,the first term on the right hand side of Equation (66),

${\lbrack \frac{(P)(S)( 10^{6} )}{nV} \rbrack ( {1 - e^{- {nt}}} )},$

will equal 0 because no additional CO₂ will have been added to theinterior of the building once the sources causing any increase of theindoor CO₂ concentration are negated. Thus, Equation (66) can besimplified to:

C _(t) ^(Inside) =C ^(Outside)+(C ₀ ^(Inside) −C ^(Outside))e^(−nt)  (67)

Second, infiltration, as represented by the number of ACH, can bedetermined by solving Equation (67) for n:

$\begin{matrix}{n = {{\ln ( \frac{C_{0}^{Inside} - C^{Outside}}{C_{t}^{Inside} - C^{Outside}} )}( \frac{1}{t} )}} & (68)\end{matrix}$

Note that the solution is only valid under conditions where t is greaterthan 0 and t is not excessively large, where C_(t)^(Inside)=C^(Outside). Last, conduction can be found by subtracting theinfiltration from the building's total thermal conductivity.

Fully Occupied Building Equilibrium

In a still further embodiment, once the number of ACH has been measuredusing Equation (68), the result can be confirmed using long-term data.The approach is to evaluate CO₂ concentrations under steady-state,fully-occupied conditions, which occurs when time t is large. Underthese conditions, Equation (66) simplifies to:

$\begin{matrix}{C_{t}^{Inside} = {\lbrack \frac{(P)(S)( 10^{6} )}{nV} \rbrack + C^{Outside}}} & (69)\end{matrix}$

Equation (69) provides an upper bound on estimated indoor CO₂concentration. Thus, ┌C_(t) ^(Inside)┐ would provide a limit to indoorCO₂ concentration beyond which C_(t) ^(Inside) would not exceed. Theequation can be used assuming that the number of people is known andremains fairly constant over time t, CO₂ addition per person can beestimated, and the volume of the building V can be determined.

Long Duration Test Validation

Equation (69) can be rearranged to calculate the number of ACH understeady-state, fully-occupied conditions if certain variables are known:

$\begin{matrix}{n = {\lbrack \frac{(P)(S)}{(V)( 10^{6} )} \rbrack ( \frac{1}{C_{t}^{Inside} - C^{Outside}} )}} & (70)\end{matrix}$

Examples are discussed infra to compare the results generated byEquations (68) and (70).

Validation

To validate the methodology, a CO₂ concentration monitoring device wasinstalled in a test house in Napa, Calif. The test house is awell-sealed, 3,000 ft² house. FIG. 23 is a graph 240 showing, by way ofexample, a time series of CO₂ concentration levels inside a test houseas measured every half-hour from Nov. 1 to Dec. 31, 2015. The x-axisrepresents the day of the month. They-axis represents CO₂ concentrationin ppm. The solid black line shows the CO₂ concentration measureddownstairs and the dotted gray line shows the CO₂ concentration measuredupstairs. One question is how many CO₂ concentration monitoring devicesare required in a building to perform this type of test. The answerdepends upon how well the CO₂ mixes throughout the building. FIG. 24 isa graph 250 showing, by way of example, the monitored CO₂ concentrationas measured in locations upstairs and downstairs in the test houseduring the two-month period. The x- and y-axes represent CO₂concentration in ppm. The solid black line shows the trend between theuncalibrated data and the dashed black line shows what the trend wouldhave been if the devices had the same calibration. The data suggeststhat, while there is a small calibration difference between the datacollected at the upstairs and downstairs locations, the half-hourmeasurements at the two different locations are linearly related to eachother most of the time with only a 5% absolute error on a half-hourbasis once the two measurements are correctly calibrated. Both sensorsprovide a good relative measurement of CO₂ concentration, which suggeststhat CO₂ mixes rather quickly throughout the building and that onedevice would have been sufficient to perform the test in this case.

FIG. 25 is a graph 260 showing, by way of example, a time series of CO₂concentration levels inside and outside the test house as measured everyhalf-hour from Nov. 1 to Dec. 31, 2015. The x-axis represents the day ofthe month. They-axis represents CO₂ concentration in ppm. The solidblack line shows the indoor CO₂ concentration, as an average of upstairsand downstairs measurements, and the dotted gray line shows theconcentration CO₂ measured in the attic, which is representative of theoutdoor CO₂ concentration. The data suggests that there was a noticeabledrop in indoor CO₂ concentration starting at around 9 am, which occurredwhen the test house's two occupants left for several days.

The absence of the occupants from the test house presented a goodopportunity to perform the short-duration empty house test to see howmany hours would be required to determine the air leakage rate in ACH.FIG. 26 is a graph showing, by way of example, a time series of CO₂concentration levels inside and outside the test house as measured everyhalf-hour for a 24-hour period on Dec. 10, 2015. The x-axis representstest duration in hours. They-axis represents CO₂ concentration in ppm.The top line shows the CO₂ concentration measured downstairs. The middleline shows the CO₂ concentration measured upstairs. The bottom lineshows the CO₂ concentration measured in the attic. The data have notbeen calibrated.

Based on the recorded data, the number of ACH was determined usingEquation (68) with CO₂ concentration graphed as a function of time. FIG.27 is a graph 280 showing, by way of example, a time series of numbersof ACH for a 24-hour period on Dec. 10, 2015. The x-axis represents testduration in hours. The y-axis represents ACH. Based on the recordeddata, the average CO₂ concentration in the attic was 483 ppm. Theinitial CO₂ concentrations upstairs and downstairs respectively were1,144 ppm and 1,297 ppm. The CO₂ concentration downstairs one hour afterthe test began was 1,247 ppm. Thus, according to Equation (68), thehouse had an ACH of 0.063 (0.063=ln((1,297−483)/(1,247−483))(1/1)) onehour after the test was initiated. While there is some variation basedon the location and duration of the test, results indicate that thehouse has an air leakage rate of about 0.06 ACH. Note that there isminimal sensitivity to outdoor CO₂ concentration. Assuming, for example,that the outdoor CO₂ concentration was a default value of 400 ppm,rather than the measured value of 483 ppm in the Attic, the house wouldhave had an ACH of 0.057 (0.057=ln((1,297−400)/(1,247−400))(1/1)) onehour after the test initiated, which still rounds up to 0.06. Thesefindings suggest that acceptable results can be obtained by performing aone-hour test and assuming a default value for outdoor CO₂concentration.

Further Validation

Equation (69) can be used to validate the accuracy of test results bycomparing actual measurements to steady state conditions. The test houseis 3,000 ft² with an average ceiling height of eight feet to create avolume of 24,000 ft³. Thus, assuming that the two occupants would emit1.42 ft³ of CO₂ per hour, the steady-state CO₂ equilibrium for equals1,483 ppm:

$\begin{matrix}{C_{t}^{Inside} = {{\lbrack \frac{(P)(S)( 10^{6} )}{nV} \rbrack + C^{Outside}} = {{497 + \lbrack \frac{( {2\mspace{14mu} {people}} )( {{0.71\frac{{ft}^{3}}{person}} - {hour}} )( 10^{6} )}{( {0.06\mspace{14mu} {air}\frac{change}{hour}} )( {24{,000\mspace{14mu} {ft}^{3}}} )} \rbrack} = {1,483}}}} & (71)\end{matrix}$

Steady state CO₂ equilibrium can be estimated for larger bodies ofpeople. FIG. 28 is a graph showing, by way of example, a steady stateCO₂ concentration levels inside and outside the test house as projectedfrom Nov. 1 to Dec. 31, 2015 for two, four, and six people. Theprojected data corresponds nicely with actual conditions. For instance,the house was occupied by two people for most of the time period underconsideration, except for the Thanksgiving and Christmas holidays, wherethe house contained up to six occupants, and the weeks after Christmas,where the house contained six occupants. The house had no externalventilation during this time.

Energy Consumption Modeling System Modeling Energy Consumption forHeating (or Cooling) on an Annual (or Periodic) Basis, as DescribedSupra with Reference BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1, and on an hourly (or interval) basis, as described suprabeginning with reference to FIG. 11, can be performed with theassistance of a computer, or through the use of hardware tailored to thepurpose. In a further embodiment, the number of ACH can be empiricallymeasured under actual operating conditions, as described supra beginningwith reference to FIG. 22, which enables infiltration and conduction tobe directly determined; this methodology can be performed with theassistance of a CO₂ concentration monitoring device and a computer, orthrough the use of hardware tailored to the purpose. FIG. 29 is a blockdiagram showing a system 310 for determining infiltration of a building313 through empirical testing using a CO₂ concentration monitoringdevice, in accordance with a further embodiment. A computer system 311,such as a personal, notebook, or tablet computer, as well as asmartphone or programmable mobile device, can be programmed to executesoftware programs 312 that operate autonomously or under user control,as provided through user interfacing means, such as a monitor, keyboard,and mouse. The computer system 311 includes hardware componentsconventionally found in a general purpose programmable computing device,such as a central processing unit, memory, input/output ports, networkinterface, and non-volatile storage, and execute the software programs312, as structured into routines, functions, and modules. In addition,other configurations of computational resources, whether provided as adedicated system or arranged in client-server or peer-to-peertopologies, and including unitary or distributed processing,communications, storage, and user interfacing, are possible.

In one embodiment, to perform the first approach, the computer system311 needs data on heating losses and heating gains, with the latterseparated into internal heating gains (occupant, electric, and solar)and auxiliary heating gains. The computer system 311 may be remotelyinterfaced with a server 320 operated by a power utility or otherutility service provider 321 over a wide area network 319, such as theInternet, from which fuel purchase data 322 can be retrieved.Optionally, the computer system 311 may also monitor electricity 314 andother metered fuel consumption, where the meter is able to externallyinterface to a remote machine, as well as monitor on-site powergeneration, such as generated by a photovoltaic system 315. Themonitored fuel consumption and power generation data can be used tocreate the electricity and heating fuel consumption data and historicalsolar resource and weather data. The computer system 311 then executes asoftware program 312 to determine annual (or periodic) heating fuelconsumption based on the empirical approach described supra withreference to BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1.

In a further embodiment, to assist with the empirical tests performed inthe second approach, the computer system 311 can be remotely interfacedto a heating source 316 and a thermometer 317 inside a building 313 thatis being analytically evaluated for thermal performance, thermal mass,effective window area, and HVAC system efficiency. In a furtherembodiment, the computer system 311 also remotely interfaces to athermometer 318 outside the building 163, or to a remote data sourcethat can provide the outdoor temperature. The computer system 311 cancontrol the heating source 316 and read temperature measurements fromthe thermometer 317 throughout the short-duration empirical tests. In afurther embodiment, a cooling source (not shown) can be used in place ofor in addition to the heating source 316. The computer system 311 thenexecutes a software program 312 to determine hourly (or interval)heating fuel consumption based on the empirical approach described suprawith reference to FIG. 11.

In a still further embodiment, a CO₂ concentration monitoring device 323monitors CO₂ concentration inside the building 313 and may also be usedto measure CO₂ concentration outside the building 313, depending uponthe testing configuration. The CO₂ concentration monitoring device 323can be remotely interfaced to a mobile device 324, such as a smartphone,or the computer system 311. The CO₂ concentration monitoring device 323monitors and records the CO₂ concentration during the test and the CO₂concentration recordings 325 are obtained by the mobile device 324. Themobile device 324 (or computer system 313) then executes an application(“App”) 326 (or program 327) to determine infiltration and conduction328 based on the empirical approach described supra with reference toFIG. 22.

Applications

The two approaches to estimating energy consumption for heating (orcooling), hourly and annual, provide a powerful set of tools that can beused in various applications. A non-exhaustive list of potentialapplications will now be discussed. Still other potential applicationsare possible.

Application to Homeowners

Both of the approaches, annual (or periodic) and hourly (or interval),reformulate fundamental building heating (and cooling) analysis in amanner that can divide a building's thermal conductivity into two parts,one part associated with the balance point resulting from internal gainsand one part associated with auxiliary heating requirements. These twoparts provide that:

-   -   Consumers can compare their house to their neighbors' houses on        both a total thermal conductivity UA^(Total) basis and on a        balance point per square foot basis. These two numbers, total        thermal conductivity UA^(Total) and balance point per square        foot, can characterize how well their house is doing compared to        their neighbors' houses. The comparison could also be performed        on a neighborhood- or city-wide basis, or between comparably        built houses in a subdivision. Other types of comparisons are        possible.    -   As strongly implied by the empirical analyses discussed supra,        heater size can be significantly reduced as the interior        temperature of a house approaches its balance point temperature.        While useful from a capital cost perspective, a heater that was        sized based on this implication may be slow to heat up the house        and could require long lead times to anticipate heating needs.        Temperature and solar forecasts can be used to operate the        heater by application of the two approaches described supra, so        as to optimize operation and minimize consumption. For example,        if the building owner or occupant knew that the sun was going to        start adding a lot of heat to the building in a few hours, he        may choose to not have the heater turn on. Alternatively, if the        consumer was using a heater with a low power rating, he would        know when to turn the heater off to achieve desired preferences.

Application to Building Shell Investment Valuation

The economic value of heating (and cooling) energy savings associatedwith any building shell improvement in any building has been shown to beindependent of building type, age, occupancy, efficiency level, usagetype, amount of internal electric gains, or amount solar gains, providedthat fuel has been consumed at some point for auxiliary heating. Asindicated by Equation (47), the only information required to calculatesavings includes the number of hours that define the winter season;average indoor temperature; average outdoor temperature; the building'sHVAC system efficiency (or coefficient of performance for heat pumpsystems); the area of the existing portion of the building to beupgraded; the R-value of the new and existing materials; and the averageprice of energy, that is, heating fuel. This finding means, for example,that a high efficiency window replacing similar low efficiency windowsin two different buildings in the same geographical location for twodifferent customer types, for instance, a residential customer versus anindustrial customer, has the same economic value, as long as the HVACsystem efficiencies and fuel prices are the same for these two differentcustomers.

This finding vastly simplifies the process of analyzing the value ofbuilding shell investments by fundamentally altering how the analysisneeds to be performed. Rather than requiring a full energy audit-styleanalysis of the building to assess any the costs and benefits of aparticular energy efficiency investment, only the investment ofinterest, the building's HVAC system efficiency, and the price and typeof fuel being saved are required.

As a result, the analysis of a building shell investment becomes muchmore like that of an appliance purchase, where the energy savings, forexample, equals the consumption of the old refrigerator minus the costof the new refrigerator, thereby avoiding the costs of a whole housebuilding analysis. Thus, a consumer can readily determine whether anacceptable return on investment will be realized in terms of costsversus likely energy savings. This result could be used in a variety ofplaces:

-   -   Direct display of economic impact in ecommerce sites. A Web        service that estimates economic value can be made available to        Web sites where consumers purchase building shell replacements.        The consumer would select the product they are purchasing, for        instance, a specific window, and would either specify the        product that they are replacing or a typical value can be        provided. This information would be submitted to the Web        service, which would then return an estimate of savings using        the input parameters described supra.    -   Tools for salespeople at retail and online establishments.    -   Tools for mobile or door-to-door sales people.    -   Tools to support energy auditors for immediate economic        assessment of audit findings. For example, a picture of a        specific portion of a house can be taken and the dollar value of        addressing problems can be attached.    -   Have a document with virtual sticky tabs that show economics of        exact value for each portion of the house. The document could be        used by energy auditors and other interested parties.    -   Available to companies interacting with new building purchasers        to interactively allow them to understand the effects of        different building choices from an economic (and environmental)        perspective using a computer program or Internet-based tool.    -   Enable real estate agents working with customers at the time of        a new home purchase to quantify the value of upgrades to the        building at the time of purchase.    -   Tools to simplify the optimization problem because most parts of        the problem are separable and simply require a rank ordering of        cost-benefit analysis of the various measures and do not require        detailed computer models that applied to specific houses.    -   The time to fix the insulation and ventilation in a homeowner's        attic is when during reroofing. This result could be integrated        into the roofing quoting tools.    -   Incorporated into a holistic zero net energy analysis computer        program or Web site to take an existing building to zero net        consumption.    -   Integration into tools for architects, builders, designers for        new construction or retrofit. Size building features or HVAC        system. More windows or less windows will affect HVAC system        size.

Application to Thermal Conductivity Analysis

A building's thermal conductivity can be characterized using onlymeasured utility billing data (natural gas and electricity consumption)and assumptions about effective window area, HVAC system efficiency andaverage indoor building temperature. This test could be used as follows:

-   -   Utilities lack direct methods to measure the energy savings        associated with building shell improvements. Use this test to        provide a method for electric utilities to validate energy        efficiency investments for their energy efficiency programs        without requiring an on-site visit or the typical detailed        energy audit. This method would help to address the measurement        and evaluation (M&E) issues currently associated with energy        efficiency programs.    -   HVAC companies could efficiently size HVAC systems based on        empirical results, rather than performing Manual J calculations        or using rules of thumb. This test could save customers money        because Manual J calculations require a detailed energy audit.        This test could also save customers capital costs since rules of        thumb typically oversize HVAC systems, particularly for        residential customers, by a significant margin.    -   A company could work with utilities (who have energy efficiency        goals) and real estate agents (who interact with customers when        the home is purchased) to identify and target inefficient homes        that could be upgraded at the time between sale and occupancy.        This approach greatly reduces the cost of the analysis, and the        unoccupied home offers an ideal time to perform upgrades without        any inconvenience to the homeowners.    -   Goals could be set for consumers to reduce a building's heating        needs to the point where a new HVAC system is avoided        altogether, thus saving the consumer a significant capital cost.

Application to Building Performance Studies

A building's performance can be fully characterized in terms of fourparameters using a suite of short-duration (several day) tests. The fourparameters include thermal conductivity, that is, heat losses, thermalmass, effective window area, and HVAC system efficiency. An assumptionis made about average indoor building temperature. These (or theprevious) characterizations could be used as follows:

-   -   Utilities could identify potential targets for building shell        investments using only utility billing data. Buildings could be        identified in a two-step process. First, thermal conductivity        can be calculated using only electric and natural gas billing        data, making the required assumptions presented supra. Buildings        that pass this screen could be the focus of a follow-up,        on-site, short-duration test.    -   The results from this test suite can be used to generate        detailed time series fuel consumption data (either natural gas        or electricity). This data can be combined with an economic        analysis tool, such as the PowerBill service        (http://www.cleanpower.com/products/powerbill/), a software        service offered by Clean Power Research, L.L.C., Napa, Calif.,        to calculate the economic impacts of the changes using detailed,        time-of-use rate structures.

Application to “Smart” Thermostat Users

The results from the short-duration tests, as described supra withreference to FIG. 4, could be combined with measured indoor buildingtemperature data collected using an Internet-accessible thermostat, suchas a Nest thermostat device or a Lyric thermostat device, cited supra,or other so-called “smart” thermostat devices, thereby avoiding havingto make assumptions about indoor building temperature. The buildingcharacterization parameters could then be combined with energyinvestment alternatives to educate consumers about the energy, economic,and environmental benefits associated with proposed purchases.

Applications of CO₂ Monitoring Device

The ability to empirically determine infiltration and conduction througha widely-available, low cost CO₂ monitoring device is particularlybeneficial to consumers, who become empowered with the knowledge of thedegree to which infiltration and conduction individually contribute toor adversely affect their home's total thermal conductivity. Thus,consumers are better able to improve total building envelope efficiencyby selecting energy efficiency investments according to their specificneeds to achieve better sealing or insulation.

A non-exhaustive list of potential applications of the CO₂ monitoringdevice will now be discussed. Still other potential applications arepossible.

-   -   Monitoring building occupancy once air leakage rate is known to        verify employee attendance; fewer CO₂ sensors would be required        than occupancy sensors because they do not require line-of-site        placement.    -   Controlling the operation of mechanical ventilation systems        based on CO₂ to save energy, rather than continuously running        heat exchangers or other ventilation devices.

While the invention has been particularly shown and described asreferenced to the embodiments thereof, those skilled in the art willunderstand that the foregoing and other changes in form and detail maybe made therein without departing from the spirit and scope.

What is claimed is:
 1. A system for monitoring occupancy of a buildingusing a tracer gas concentration monitoring device, comprising: a tracergas concentration monitoring device provided inside a building undertest and operable to determine and record an initial tracer gasconcentration, and further operable to measure and record further tracergas concentrations inside the building subsequent to an increase intracer gas concentration over the initial tracer gas concentration and anegation of sources causing the increase in the tracer gas concentrationinside the building until the further tracer gas concentrationsstabilize, and to record additional tracer gas concentrations followingthe stabilization; a storage, comprising: a baseline tracer gasconcentration applicable to outside the building; a total thermalconductivity of the building; a computer processor interfaced to thestorage and configured to execute code, the code comprising: aninfiltration module configured to determine infiltration of the buildingbased on a number of air changes as a function of the difference of theinitial tracer gas concentration less the baseline tracer gasconcentration over one or more of the further tracer gas concentrationat a given time less the baseline tracer gas concentration and the giventime; a conduction module configured to determine conduction of thebuilding as the difference of the total thermal conductivity less theinfiltration of the building, wherein at least one improvement to ashell of the building is performed based on the infiltration and theconduction; and a monitoring module configured to monitor occupancy ofthe building based on the determined infiltration and the additionaltracer gas concentrations.
 2. A system according to claim 1, furthercomprising: a further tracer gas concentration monitoring device checkedagainst the tracer gas concentration device and configured to measurethe baseline tracer gas concentration applicable to outside thebuilding.
 3. A system according to claim 1, further comprising: acomparison module configured to compare the initial tracer gasconcentration to the baseline tracer gas concentration; a disparitymodule configured to identify a disparity exceeding a threshold betweenthe initial tracer gas concentration and the baseline tracer gasconcentration based on the comparison, wherein a source of the disparityis removed from the building prior to the recordation of the furthertracer gas concentrations.
 4. A system according to claim 1, wherein thestabilization comprises at least one of one of the tracer gasconcentrations substantially equaling the baseline tracer gasconcentration and a plurality of the further gas concentrations beingthe same.
 5. A system according to claim 1, wherein a time from thebeginning of the negation to the stabilization is one hour.
 6. A systemaccording to claim 1, wherein the computer processor is remotelyinterfaced to the tracer gas concentration monitoring device.
 7. Asystem according to claim 6, wherein the computer processor is comprisedin a mobile phone.
 8. A system according to claim 1, wherein the tracergas is CO₂.
 9. A system according to claim 1, wherein the number of airchanges n is determined in accordance with:$n = {{\ln ( \frac{C_{0}^{Inside} - C^{Outside}}{C_{t}^{Inside} - C^{Outside}} )}( \frac{1}{t} )}$where C₀ ^(Inside) is the initial tracer gas concentration; C_(t)^(Inside) is the further tracer gas concentration at time t; C^(Outside)is the baseline tracer gas concentration; and t is time expressed inhours.
 10. A system according to claim 1, further comprising: theinfiltration module further configured to determine the infiltration ofthe building based on a number of air changes as a function ofsteady-state, fully-occupied conditions inside the building.
 11. Asystem for controlling ventilation of a building through using a tracergas concentration monitoring device, comprising: a tracer gasconcentration monitoring device provided inside a building under testand operable to determine and record an initial tracer gasconcentration, and further operable to measure and record further tracergas concentrations inside the building subsequent to an increase intracer gas concentration over the initial tracer gas concentration and anegation of sources causing the increase in the tracer gas concentrationinside the building until the further tracer gas concentrationsstabilize, and to record additional tracer gas concentrations followingthe stabilization; a storage, comprising: a baseline tracer gasconcentration applicable to outside the building; a total thermalconductivity of the building; a computer processor interfaced to thestorage and configured to execute code, the code comprising: aninfiltration module configured to determine infiltration of the buildingbased on a number of air changes as a function of the difference of theinitial tracer gas concentration less the baseline tracer gasconcentration over one or more of the further tracer gas concentrationat a given time less the baseline tracer gas concentration and the giventime; a conduction module configured to determine conduction of thebuilding as the difference of the total thermal conductivity less theinfiltration of the building, wherein at least one improvement to ashell of the building is performed based on the infiltration and theconduction; and a control module configured to control a mechanicalventilation system of the building based on the determined infiltrationand the additional tracer gas concentrations.
 12. A system according toclaim 11, further comprising: a further tracer gas concentrationmonitoring device checked against the tracer gas concentration deviceand configured to measure the baseline tracer gas concentrationapplicable to outside the building.
 13. A system according to claim 11,further comprising: a comparison module configured to compare theinitial tracer gas concentration to the baseline tracer gasconcentration; a disparity module configured to identify a disparityexceeding a threshold between the initial tracer gas concentration andthe baseline tracer gas concentration based on the comparison, wherein asource of the disparity is removed from the building prior to therecordation of the further tracer gas concentrations.
 14. A systemaccording to claim 11, wherein the stabilization comprises at least oneof one of the tracer gas concentrations substantially equaling thebaseline tracer gas concentration and a plurality of the further gasconcentrations being the same.
 15. A system according to claim 11,wherein a time from the beginning of the negation to the stabilizationis one hour.
 16. A system according to claim 11, wherein the computerprocessor is remotely interfaced to the tracer gas concentrationmonitoring device.
 17. A system according to claim 16, wherein thecomputer processor is comprised in a mobile phone.
 18. A systemaccording to claim 11, wherein the tracer gas is CO₂.
 19. A systemaccording to claim 11, wherein the number of air changes n is determinedin accordance with:$n = {{\ln ( \frac{C_{0}^{Inside} - C^{Outside}}{C_{t}^{Inside} - C^{Outside}} )}( \frac{1}{t} )}$where C₀ ^(Inside) is the initial tracer gas concentration; C_(t)^(Inside) is the further tracer gas concentration at time t; C^(Outside)is the baseline tracer gas concentration; and t is time expressed inhours.
 20. A system according to claim 11, further comprising: theinfiltration module further configured to determine the infiltration ofthe building based on a number of air changes as a function ofsteady-state, fully-occupied conditions inside the building.